This document introduces product measures and establishes foundational results about them. It defines the Cartesian product of two measurable spaces and shows how to define a measure on the product space that is the "product" of measures on the factor spaces. The key results are:
1) Tonelli's Theorem, which states that for a non-negative measurable function on a product space with σ-finite factors, the integral over the product equals the iterated integrals over the factors.
2) Fubini's Theorem, which extends this to integrable functions that may be positive or negative, showing the iterated integrals are equal almost everywhere.
3) Definitions of sections of sets and functions that are used to relate
This document defines key concepts in measure theory and integration, including:
(1) A σ-algebra is a collection of subsets of a set X that is closed under complement and countable unions. A measurable space is a set X equipped with a σ-algebra.
(2) A measure on a measurable space assigns a value in [0,∞] to elements of the σ-algebra in a countably additive way. A measure space consists of a measurable space equipped with a measure.
(3) Examples of measure spaces include Lebesgue measure on R, counting measure, and Dirac measure. Properties of measures like monotonicity and limits of sequences are proved.
Real Analysis II (Measure Theory) NotesPrakash Dabhi
This document contains an outline for a course on measure theory and integration. It discusses the following topics:
1. Measure spaces, σ-algebras, measurable functions, integration, convergence theorems.
2. Signed measures, Hahn decomposition, Jordan decomposition, Lebesgue decomposition theorem, Radon-Nikodym theorem.
3. Cumulative distribution functions, Lp spaces, Holder's inequality, Minkowski inequality, density in Lp spaces.
4. Caratheodory's extension theorem, product measures, Fubini's theorem, Tonelli's theorem, regularity of measures.
It lists reference books and provides an overview of the content to be covered in each unit of
Measure Theory and important points with bookletNaeemAhmad289736
The document provides a summary of key concepts in measure theory and Lebesgue integration. In 3 sentences: It defines topologies and σ-algebras, and describes measurable functions and spaces. It introduces measures, including positive measures and the Lebesgue measure. It covers integrals of simple and Lebesgue integrable functions, and theorems like monotone convergence, dominated convergence, and Fubini's theorem.
This document defines key concepts in measure theory and integration, including σ-algebras, measurable spaces, measures, measure spaces, properties of measures (monotonicity, countable additivity), completion of measure spaces, and locally measurable sets. It provides examples of common measure spaces such as Lebesgue measure on the real line. The document establishes several important lemmas about measures, such as monotonicity, countable subadditivity, and limits of increasing/decreasing sequences.
On Extendable Sets in the Reals (R) With Application to the Lyapunov Stabilit...BRNSS Publication Hub
This document summarizes a research article that defines extendable sets in the real numbers (R) and applies this concept to the Lyapunov stability comparison principle of ordinary differential equations. It begins with the author's own definition of extension on R and a basic result called the basic extension fact for R. It then reviews existing definitions and theorems on extension, including Urysohn's lemma and Tietze's extension theorem. The document concludes by extensively applying these results to prove some important results relating to the comparison principle of Lyapunov stability theory in ordinary differential equations.
The document contains proofs of various claims about continuous functions between metric spaces. It begins by proving that if a function f is continuous on closed subsets A and B of a metric space E whose union is E, then f is continuous on E (Problem 3). It then proves similar claims about continuity of nondecreasing functions between open intervals in R (Problem 4) and about a function's oscillation and continuity (Problem 5). The document proves several other properties of continuous functions.
MA500-2: Topological Structures 2016
Aisling McCluskey, Daron Anderson
[email protected], [email protected]
Contents
0 Preliminaries 2
1 Topological Groups 8
2 Morphisms and Isomorphisms 15
3 The Second Isomorphism Theorem 27
4 Topological Vector Spaces 42
5 The Cayley-Hamilton Theorem 43
6 The Arzelà-Ascoli theorem 44
7 Tychonoff ’s Theorem if Time Permits 45
Continuous assessment 30%; final examination 70%. There will be a weekly
workshop led by Daron during which there will be an opportunity to boost
continuous assessment marks based upon workshop participation as outlined in
class.
This module is self-contained; the notes provided shall form the module text.
Due to the broad range of topics introduced, there is no recommended text.
However General Topology by R. Engelking is a graduate-level text which has
relevant sections within it. Also Undergraduate Topology: a working textbook by
McCluskey and McMaster is a useful revision text. As usual, in-class discussion
will supplement the formal notes.
1
0 PRELIMINARIES
0 Preliminaries
Reminder 0.1. A topology τ on the set X is a family of subsets of X, called
the τ-open sets, satisfying the three axioms.
(1) Both sets X and ∅ are τ-open
(2) The union of any subfamily is again a τ-open set
(3) The intersection of any two τ-open sets is again a τ-open set
We refer to (X,τ) as a topological space. Where there is no danger of ambi-
guity, we suppress reference to the symbol denoting the topology (in this case,
τ) and simply refer to X as a topological space and to the elements of τ as its
open sets. By a closed set we mean one whose complement is open.
Reminder 0.2. A metric on the set X is a function d: X×X → R satisfying
the five axioms.
(1) d(x,y) ≥ 0 for all x,y ∈ X
(2) d(x,y) = d(y,x) for x,y ∈ X
(3) d(x,x) = 0 for every x ∈ X
(4) d(x,y) = 0 implies x = y
(5) d(x,z) ≤ d(x,y) + d(y,z) for all x,y,z ∈ X
Axiom (5) is often called the triangle inequality.
Definition 0.3. If d′ : X × X → R satisfies axioms (1), (2), (3) and (5) but
maybe not (4) then we call it a pseudo-metric.
Reminder 0.4. Every metric on X induces a topology on X, called the metric
topology. We define an open ball to be a set of the form
B(x,r) = {y ∈ X : d(x,y) < r}
for any x ∈ X and r > 0. Then a subset G of X is defined to be open (wrt the
metric topology) if for each x ∈ G, there is r > 0 such that B(x,r) ⊂ G. Thus
open sets are arbitrary unions of open balls.
Topological Structures 2016 2 Version 0.15
0 PRELIMINARIES
The definition of the metric topology makes just as much sense when we are
working with a pseudo-metric. Open balls are defined in the same manner, and
the open sets are exactly the unions of open balls. Pseudo-metric topologies are
often neglected because they do not have the nice property of being Hausdorff.
Reminder 0.5. Suppose f : X → Y is a function between the topological
spaces X and Y . We say f is continuous to mean that whenever U is open in
Y ...
This document presents research on extendable sets in the real numbers (R) and their application to the Lyapunov stability comparison principle of ordinary differential equations. It begins with definitions of the real numbers and extendable sets. It then reviews existing definitions of extension, including Urysohn's lemma and Tietze extension theorem. The main result proved is that every compact subset of R is extendable, while non-compact subsets are not. It concludes by extensively applying these results to prove important theorems regarding the comparison principle of Lyapunov stability theory in ordinary differential equations.
This document defines key concepts in measure theory and integration, including:
(1) A σ-algebra is a collection of subsets of a set X that is closed under complement and countable unions. A measurable space is a set X equipped with a σ-algebra.
(2) A measure on a measurable space assigns a value in [0,∞] to elements of the σ-algebra in a countably additive way. A measure space consists of a measurable space equipped with a measure.
(3) Examples of measure spaces include Lebesgue measure on R, counting measure, and Dirac measure. Properties of measures like monotonicity and limits of sequences are proved.
Real Analysis II (Measure Theory) NotesPrakash Dabhi
This document contains an outline for a course on measure theory and integration. It discusses the following topics:
1. Measure spaces, σ-algebras, measurable functions, integration, convergence theorems.
2. Signed measures, Hahn decomposition, Jordan decomposition, Lebesgue decomposition theorem, Radon-Nikodym theorem.
3. Cumulative distribution functions, Lp spaces, Holder's inequality, Minkowski inequality, density in Lp spaces.
4. Caratheodory's extension theorem, product measures, Fubini's theorem, Tonelli's theorem, regularity of measures.
It lists reference books and provides an overview of the content to be covered in each unit of
Measure Theory and important points with bookletNaeemAhmad289736
The document provides a summary of key concepts in measure theory and Lebesgue integration. In 3 sentences: It defines topologies and σ-algebras, and describes measurable functions and spaces. It introduces measures, including positive measures and the Lebesgue measure. It covers integrals of simple and Lebesgue integrable functions, and theorems like monotone convergence, dominated convergence, and Fubini's theorem.
This document defines key concepts in measure theory and integration, including σ-algebras, measurable spaces, measures, measure spaces, properties of measures (monotonicity, countable additivity), completion of measure spaces, and locally measurable sets. It provides examples of common measure spaces such as Lebesgue measure on the real line. The document establishes several important lemmas about measures, such as monotonicity, countable subadditivity, and limits of increasing/decreasing sequences.
On Extendable Sets in the Reals (R) With Application to the Lyapunov Stabilit...BRNSS Publication Hub
This document summarizes a research article that defines extendable sets in the real numbers (R) and applies this concept to the Lyapunov stability comparison principle of ordinary differential equations. It begins with the author's own definition of extension on R and a basic result called the basic extension fact for R. It then reviews existing definitions and theorems on extension, including Urysohn's lemma and Tietze's extension theorem. The document concludes by extensively applying these results to prove some important results relating to the comparison principle of Lyapunov stability theory in ordinary differential equations.
The document contains proofs of various claims about continuous functions between metric spaces. It begins by proving that if a function f is continuous on closed subsets A and B of a metric space E whose union is E, then f is continuous on E (Problem 3). It then proves similar claims about continuity of nondecreasing functions between open intervals in R (Problem 4) and about a function's oscillation and continuity (Problem 5). The document proves several other properties of continuous functions.
MA500-2: Topological Structures 2016
Aisling McCluskey, Daron Anderson
[email protected], [email protected]
Contents
0 Preliminaries 2
1 Topological Groups 8
2 Morphisms and Isomorphisms 15
3 The Second Isomorphism Theorem 27
4 Topological Vector Spaces 42
5 The Cayley-Hamilton Theorem 43
6 The Arzelà-Ascoli theorem 44
7 Tychonoff ’s Theorem if Time Permits 45
Continuous assessment 30%; final examination 70%. There will be a weekly
workshop led by Daron during which there will be an opportunity to boost
continuous assessment marks based upon workshop participation as outlined in
class.
This module is self-contained; the notes provided shall form the module text.
Due to the broad range of topics introduced, there is no recommended text.
However General Topology by R. Engelking is a graduate-level text which has
relevant sections within it. Also Undergraduate Topology: a working textbook by
McCluskey and McMaster is a useful revision text. As usual, in-class discussion
will supplement the formal notes.
1
0 PRELIMINARIES
0 Preliminaries
Reminder 0.1. A topology τ on the set X is a family of subsets of X, called
the τ-open sets, satisfying the three axioms.
(1) Both sets X and ∅ are τ-open
(2) The union of any subfamily is again a τ-open set
(3) The intersection of any two τ-open sets is again a τ-open set
We refer to (X,τ) as a topological space. Where there is no danger of ambi-
guity, we suppress reference to the symbol denoting the topology (in this case,
τ) and simply refer to X as a topological space and to the elements of τ as its
open sets. By a closed set we mean one whose complement is open.
Reminder 0.2. A metric on the set X is a function d: X×X → R satisfying
the five axioms.
(1) d(x,y) ≥ 0 for all x,y ∈ X
(2) d(x,y) = d(y,x) for x,y ∈ X
(3) d(x,x) = 0 for every x ∈ X
(4) d(x,y) = 0 implies x = y
(5) d(x,z) ≤ d(x,y) + d(y,z) for all x,y,z ∈ X
Axiom (5) is often called the triangle inequality.
Definition 0.3. If d′ : X × X → R satisfies axioms (1), (2), (3) and (5) but
maybe not (4) then we call it a pseudo-metric.
Reminder 0.4. Every metric on X induces a topology on X, called the metric
topology. We define an open ball to be a set of the form
B(x,r) = {y ∈ X : d(x,y) < r}
for any x ∈ X and r > 0. Then a subset G of X is defined to be open (wrt the
metric topology) if for each x ∈ G, there is r > 0 such that B(x,r) ⊂ G. Thus
open sets are arbitrary unions of open balls.
Topological Structures 2016 2 Version 0.15
0 PRELIMINARIES
The definition of the metric topology makes just as much sense when we are
working with a pseudo-metric. Open balls are defined in the same manner, and
the open sets are exactly the unions of open balls. Pseudo-metric topologies are
often neglected because they do not have the nice property of being Hausdorff.
Reminder 0.5. Suppose f : X → Y is a function between the topological
spaces X and Y . We say f is continuous to mean that whenever U is open in
Y ...
This document presents research on extendable sets in the real numbers (R) and their application to the Lyapunov stability comparison principle of ordinary differential equations. It begins with definitions of the real numbers and extendable sets. It then reviews existing definitions of extension, including Urysohn's lemma and Tietze extension theorem. The main result proved is that every compact subset of R is extendable, while non-compact subsets are not. It concludes by extensively applying these results to prove important theorems regarding the comparison principle of Lyapunov stability theory in ordinary differential equations.
This document summarizes a research article that defines extendable sets in the real numbers (R) and applies this concept to proofs involving the Lyapunov stability comparison principle of ordinary differential equations. It begins with the author's own definition of an extension on R and a basic result called the basic extension fact for R. Existing definitions and theorems on extension, such as Urysohn's lemma and Tietze's extension theorem, are then reviewed. The document concludes by extensively applying these concepts to resolve proofs involving the comparison principle of Lyapunov stability theory.
This document provides an overview and proofs of several theorems related to the Hahn-Banach theorem. It begins with an introduction to linear functionals and the Hahn-Banach theorem. It then presents two main theorems - the Hahn-Banach theorem and the topological Hahn-Banach theorem. The document provides proofs of these theorems and several related theorems using the Hahn-Banach extension lemma. It also discusses consequences of the Hahn-Banach extension form and provides proofs of the theorems using the lemma.
On Analytic Review of Hahn–Banach Extension Results with Some GeneralizationsBRNSS Publication Hub
The useful Hahn–Banach theorem in functional analysis has significantly been in use for many years ago. At this point in time, we discover that its domain and range of existence can be extended point wisely so as to secure a wider range of extendibility. In achieving this, we initially reviewed the existing traditional Hahn–Banach extension theorem, before we carefully and successfully used it to generate the finite extension form as in main results of section three.
This document provides an overview of preliminary topological concepts needed for applied mathematics. It defines topological spaces and metric spaces, and introduces key topological notions like open and closed sets, bases for topologies, convergence of sequences, accumulation points, interior and closure of sets, and dense sets. Metric spaces are shown to induce a natural topological structure, though not all topologies come from a metric. Examples are provided to illustrate various definitions and properties.
This document discusses various methods for solving first order differential equations, including:
1. Variable separable methods where the equation can be written as a function of x multiplied by a function of y.
2. Homogeneous equations where both sides are homogeneous functions of the same degree.
3. Exact equations where there exists an integrating factor.
4. Equations that can be transformed to an exact or separable form through substitution.
5. Linear equations that can be solved using an integrating factor that is a function of x.
This document summarizes part of a paper by D.G. Northcott on the notion of a first neighbourhood ring, with an application to the Af+Bφ theorem. It introduces the concept of a first neighbourhood ring and superficial elements. The key points are:
1) It defines a superficial element as one where amv+s = mv+s for large v, generalizing the definition to allow zero divisors.
2) It proves three properties are equivalent for an element a to be superficial: the form ideal no:(φ) is the isolated component no, amv = mv+s for large v, and nv+s:(a) = mv for large v.
3) It shows
Vector measures and classical disjointification methodsesasancpe
1. The document discusses applying classical disjointification methods (Bessaga-Pelczynski and Kadecs-Pelczynski) to spaces of p-integrable functions with respect to vector measures.
2. These methods allow working with orthogonality notions in the range space and analyzing disjoint functions.
3. Combining the results provides tools to analyze the structure of subspaces in these spaces of p-integrable functions.
A Komlo ́sTheorem for general Banach lattices of measurable functionsesasancpe
This document summarizes research on generalizing the Koml ́os theorem to Banach lattices of measurable functions. Specifically:
1) It generalizes previous results showing the Koml ́os theorem holds for Banach function spaces to more general spaces of integrable functions with respect to vector measures.
2) It proves the Fatou and Koml ́os properties are equivalent for sublattices of spaces L1(ν) with the Fatou property and for L1w(ν) spaces.
3) It provides an example of a Banach lattice of measurable functions that has the Fatou property but does not satisfy the Koml ́os theorem, showing the properties are not always equivalent
This document defines metric spaces and discusses their basic properties. It begins by defining what a metric is and what constitutes a metric space. It provides some basic examples of metrics, such as the discrete metric and p-norm metrics. It then discusses metric topologies, defining open and closed balls and showing that the collection of open sets forms a topology. It also introduces the concept of topologically equivalent metrics.
This document provides an introduction to uniform spaces. It begins with definitions of uniformities, entourages, and basic properties of uniform spaces such as being Hausdorff. It introduces the concept of a uniform topology induced by a uniformity. Metrics and metric spaces are discussed in the context of uniformities. A key result is that a uniform space with a countable base is metrizable, meaning it can be given a metric that induces the same uniformity. The document provides proofs of basic results about uniform spaces.
This document presents theorems and results regarding hypercyclic operators on the space Hbc(E), where E is a Banach space. Theorem 3.1 shows that the collection of functions {eφ : φ ∈ E*} forms an independently linear subset of Hbc(E). Theorem 3.2 proves that the span of {eφ : φ ∈ U} is dense in Hbc(E), where U is an open subset of E*. Theorem 3.3 demonstrates that if φ is an entire function of exponential type, then the operator φα(D) is hypercyclic on Hbc(E). The document also provides two corollaries: if E has a separable dual, then
1. The document discusses the relationship between the Fourier series of a function and the Fourier series of its integral and derivative. It shows that integrating the Fourier series of a function term-by-term yields the Fourier series of its integral.
2. An example is provided to illustrate integrating the Fourier series term-by-term to evaluate a definite integral.
3. The document also proves an isoperimetric inequality stating that for a closed curve in the plane, the ratio of its perimeter to the square root of its enclosed area is always greater than or equal to 2π, with equality holding only for circles.
The document discusses concepts related to partial differentiation and its applications. It covers topics like tangent planes, linear approximations, differentials, Taylor expansions, maxima and minima problems, and the Lagrange method. Specifically, it defines the tangent plane to a surface at a point using partial derivatives, describes how to find the linear approximation of functions, and explains how to find maximum and minimum values of functions using critical points and the second derivative test.
Fixed Point Results In Fuzzy Menger Space With Common Property (E.A.)IJERA Editor
This paper presents some common fixed point theorems for weakly compatible mappings via an implicit relation in Fuzzy Menger spaces satisfying the common property (E.A)
(i) The document provides solutions to exercises from Chapter 1 of Atiyah and MacDonald's Introduction to Commutative Algebra.
(ii) It works through proofs for various statements about rings, ideals, nilpotent and Jacobson radicals, and the prime and Zariski spectra of rings.
(iii) The solutions cover topics such as when a sum of a nilpotent element and unit is a unit, when a polynomial is a unit or zero divisor based on its coefficients, and properties of the prime and Zariski topologies on the prime spectrum of a ring.
This document provides an overview and definitions related to functional analysis and Banach spaces. It discusses:
1) The definition of a Banach space as a complete normed linear space where every Cauchy sequence converges.
2) Examples of Banach spaces including l^p spaces, C(X) for compact X, C_b(X) for any topological space X, and C^k([a,b]).
3) Measure spaces and the definition of measurable functions on a measure space. It notes the closure properties of measurable functions under scalar multiplication and (sometimes) addition.
This document discusses the convexity of the set of k-admissible functions on a compact Kähler manifold. It begins by introducing k-admissible functions and some necessary convex analysis concepts. It then proves four main results: 1) the log of elementary symmetric functions of eigenvalues is a convex function, 2) the set of matrices with eigenvalues in a convex set is convex, 3) certain functions of eigenvalues are convex, and 4) the set of k-admissible functions is convex. It uses results on conjugation of spectral functions from convex analysis to prove these results. The proofs rely on properties of convex, lower semicontinuous functions and indicator functions.
The International Journal of Engineering & Science is aimed at providing a platform for researchers, engineers, scientists, or educators to publish their original research results, to exchange new ideas, to disseminate information in innovative designs, engineering experiences and technological skills. It is also the Journal's objective to promote engineering and technology education. All papers submitted to the Journal will be blind peer-reviewed. Only original articles will be published.
The papers for publication in The International Journal of Engineering& Science are selected through rigorous peer reviews to ensure originality, timeliness, relevance, and readability.
Theoretical work submitted to the Journal should be original in its motivation or modeling structure. Empirical analysis should be based on a theoretical framework and should be capable of replication. It is expected that all materials required for replication (including computer programs and data sets) should be available upon request to the authors.
Lecture13p.pdf.pdfThedeepness of freedom are threevalues.docxcroysierkathey
Lecture13p.pdf.pdf
Thedeepness of freedom are threevalues at thenude
functional Notconforming
patrtaf.us
vi sci x
I beease
ittouch
41 u VCsci inhalfedgeL
U VCI't x
Since u CPz are sci sc 7 that it have 3 Zeusunless e o
E is P anisolvent it forgiven I lop C P s t 4 p di
same
degree
y i l N Yi C E
Sabi n ofsystem YCp g
This is equivalent to say theonlypolynomial C PthetinterpolateZero
data Yifp o is the Zeno poly
vcpi.POTFF.gg In Edem
e e
I CRIvalue VCR Ca Ya
metfunctor
p
E3
pjJ Chip J
Shun E is p unisolvent
Y Cul VCR7 0
Xz V UCR o
rf VI UCB 0
Then over the edge PP we hone
C P havingtworootsPR
D This implies we 0
If e consider the other to edges G e b thesame
argument we can see Eo tht means W Lv o hersonly trivial
Solution
Then Yi CUI Ri for any Xi
E is P unisowent
y
csiy
Ya
f P Y cnn.PT
III Ldj Pg I Pre 2 ily
a PyO ein a
451214 7 f p i y g d CP f
ftp b f CRI I B so
fickle Cps O y Cp 7 L
Escaple5 in lectureto
Lecture_03_S08.pdf
LECTURE # 3:
ABSTRACT RITZ-GALERKIN METHOD
MATH610: NUMERICAL METHODS FOR PDES:
RAYTCHO LAZAROV
1. Variational Formulation
In the previous lecture we have introduced the following space of functions
defined on (0, 1):
(1)
V =
v :
v(x) is continuous function on (0, 1);
v′(x) exists in generalized sense and in L2(0, 1);
v(0) = v(1) = 0
:= H10 (0, 1)
and equipped it with the L2 and H1 norms
‖v‖ = (v,v)1/2 and ‖v‖V = (v,v)
1/2
V =
(∫ 1
0
(u′2 + u2)dx
)1
2
.
We also introduced the following variational and minimization problems:
(V ) find u ∈ V such that a(u,v) = L(v), ∀ v ∈ V,
(M) find u ∈ V such that F(u) ≤ F(v), ∀ v ∈ V,
where a(u,v) is a bilinear form that is symmetric, coercive and contin-
uous on V and L(v) is continuous on V and F(v) = 1
2
a(u,u) −L(v).
As an example we can take
a(u,v) ≡
∫ 1
0
(k(x)u′v′ + q(x)uv) dx and L(v) ≡
∫ 1
0
f(x)v dx.
Here we have assumed that there are positive constants k0, k1, M such that
(2) k1 ≥ k(x) ≥ k0 > 0, M ≥ q(x) ≥ 0, f ∈ L2(0, 1).
These are sufficient for the symmetry, coercivity and continuity of the
bilinear form a(., .) and the continuity of the linear form L(v).
The proof of these properties follows from the following theorem:
Theorem 1. Let u ∈ V ≡ H10 (0, 1). Then the following inequalities are
valid:
(3)
|u(x)|2 ≤ C1
∫ 1
0
(u′(x))2dx for any x ∈ (0, 1),∫ 1
0
u2(x)dx ≤ C0
∫ 1
0
(u′(x))2dx.
with constants C0 and C1 that are independent of u.
1
2 MATH610: NUMERICAL METHODS FOR PDES: RAYTCHO LAZAROV
Proof: We give two proofs. The simple one proves the above inequali-
ties with C0 = 1/2 and C1 = 1. The better proof establishes the above
inequalities with C0 = 1/6 and C1 = 1/4.
Indeed, for any x ∈ (0, 1) we have:
u(x) = u(0) +
∫ x
0
u′(s)ds.
Since u ∈ H10 (0, 1) then u(0) = 0. We square this equality and apply
Cauchy-Swartz inequality:
(4) |u(x)|2 =
∣∣∣∫ x
0
u′(s)ds
∣∣∣2 ≤ ∫ x
0
1ds
∫ x
0
(u′(s))2ds ≤ x
∫ x
0
(u′(s))2ds.
Taking the maximal value of x on the right hand side of this inequality
w ...
Lecture13p.pdf.pdfThedeepness of freedom are threevalues.docxjeremylockett77
The document discusses the abstract Ritz-Galerkin method for numerical solution of partial differential equations.
It begins by introducing the variational formulation and defining the function spaces used. It then describes the abstract form of the Ritz-Galerkin method, which finds an approximate solution in a finite dimensional subspace.
Specific examples are given to illustrate the method, including choice of basis functions for different boundary conditions. Basis functions for linear finite elements are defined on a partition of the domain into subintervals. The Ritz-Galerkin method results in a system of linear equations that can be solved for the coefficients of the approximate solution.
This document summarizes a research article that defines extendable sets in the real numbers (R) and applies this concept to proofs involving the Lyapunov stability comparison principle of ordinary differential equations. It begins with the author's own definition of an extension on R and a basic result called the basic extension fact for R. Existing definitions and theorems on extension, such as Urysohn's lemma and Tietze's extension theorem, are then reviewed. The document concludes by extensively applying these concepts to resolve proofs involving the comparison principle of Lyapunov stability theory.
This document provides an overview and proofs of several theorems related to the Hahn-Banach theorem. It begins with an introduction to linear functionals and the Hahn-Banach theorem. It then presents two main theorems - the Hahn-Banach theorem and the topological Hahn-Banach theorem. The document provides proofs of these theorems and several related theorems using the Hahn-Banach extension lemma. It also discusses consequences of the Hahn-Banach extension form and provides proofs of the theorems using the lemma.
On Analytic Review of Hahn–Banach Extension Results with Some GeneralizationsBRNSS Publication Hub
The useful Hahn–Banach theorem in functional analysis has significantly been in use for many years ago. At this point in time, we discover that its domain and range of existence can be extended point wisely so as to secure a wider range of extendibility. In achieving this, we initially reviewed the existing traditional Hahn–Banach extension theorem, before we carefully and successfully used it to generate the finite extension form as in main results of section three.
This document provides an overview of preliminary topological concepts needed for applied mathematics. It defines topological spaces and metric spaces, and introduces key topological notions like open and closed sets, bases for topologies, convergence of sequences, accumulation points, interior and closure of sets, and dense sets. Metric spaces are shown to induce a natural topological structure, though not all topologies come from a metric. Examples are provided to illustrate various definitions and properties.
This document discusses various methods for solving first order differential equations, including:
1. Variable separable methods where the equation can be written as a function of x multiplied by a function of y.
2. Homogeneous equations where both sides are homogeneous functions of the same degree.
3. Exact equations where there exists an integrating factor.
4. Equations that can be transformed to an exact or separable form through substitution.
5. Linear equations that can be solved using an integrating factor that is a function of x.
This document summarizes part of a paper by D.G. Northcott on the notion of a first neighbourhood ring, with an application to the Af+Bφ theorem. It introduces the concept of a first neighbourhood ring and superficial elements. The key points are:
1) It defines a superficial element as one where amv+s = mv+s for large v, generalizing the definition to allow zero divisors.
2) It proves three properties are equivalent for an element a to be superficial: the form ideal no:(φ) is the isolated component no, amv = mv+s for large v, and nv+s:(a) = mv for large v.
3) It shows
Vector measures and classical disjointification methodsesasancpe
1. The document discusses applying classical disjointification methods (Bessaga-Pelczynski and Kadecs-Pelczynski) to spaces of p-integrable functions with respect to vector measures.
2. These methods allow working with orthogonality notions in the range space and analyzing disjoint functions.
3. Combining the results provides tools to analyze the structure of subspaces in these spaces of p-integrable functions.
A Komlo ́sTheorem for general Banach lattices of measurable functionsesasancpe
This document summarizes research on generalizing the Koml ́os theorem to Banach lattices of measurable functions. Specifically:
1) It generalizes previous results showing the Koml ́os theorem holds for Banach function spaces to more general spaces of integrable functions with respect to vector measures.
2) It proves the Fatou and Koml ́os properties are equivalent for sublattices of spaces L1(ν) with the Fatou property and for L1w(ν) spaces.
3) It provides an example of a Banach lattice of measurable functions that has the Fatou property but does not satisfy the Koml ́os theorem, showing the properties are not always equivalent
This document defines metric spaces and discusses their basic properties. It begins by defining what a metric is and what constitutes a metric space. It provides some basic examples of metrics, such as the discrete metric and p-norm metrics. It then discusses metric topologies, defining open and closed balls and showing that the collection of open sets forms a topology. It also introduces the concept of topologically equivalent metrics.
This document provides an introduction to uniform spaces. It begins with definitions of uniformities, entourages, and basic properties of uniform spaces such as being Hausdorff. It introduces the concept of a uniform topology induced by a uniformity. Metrics and metric spaces are discussed in the context of uniformities. A key result is that a uniform space with a countable base is metrizable, meaning it can be given a metric that induces the same uniformity. The document provides proofs of basic results about uniform spaces.
This document presents theorems and results regarding hypercyclic operators on the space Hbc(E), where E is a Banach space. Theorem 3.1 shows that the collection of functions {eφ : φ ∈ E*} forms an independently linear subset of Hbc(E). Theorem 3.2 proves that the span of {eφ : φ ∈ U} is dense in Hbc(E), where U is an open subset of E*. Theorem 3.3 demonstrates that if φ is an entire function of exponential type, then the operator φα(D) is hypercyclic on Hbc(E). The document also provides two corollaries: if E has a separable dual, then
1. The document discusses the relationship between the Fourier series of a function and the Fourier series of its integral and derivative. It shows that integrating the Fourier series of a function term-by-term yields the Fourier series of its integral.
2. An example is provided to illustrate integrating the Fourier series term-by-term to evaluate a definite integral.
3. The document also proves an isoperimetric inequality stating that for a closed curve in the plane, the ratio of its perimeter to the square root of its enclosed area is always greater than or equal to 2π, with equality holding only for circles.
The document discusses concepts related to partial differentiation and its applications. It covers topics like tangent planes, linear approximations, differentials, Taylor expansions, maxima and minima problems, and the Lagrange method. Specifically, it defines the tangent plane to a surface at a point using partial derivatives, describes how to find the linear approximation of functions, and explains how to find maximum and minimum values of functions using critical points and the second derivative test.
Fixed Point Results In Fuzzy Menger Space With Common Property (E.A.)IJERA Editor
This paper presents some common fixed point theorems for weakly compatible mappings via an implicit relation in Fuzzy Menger spaces satisfying the common property (E.A)
(i) The document provides solutions to exercises from Chapter 1 of Atiyah and MacDonald's Introduction to Commutative Algebra.
(ii) It works through proofs for various statements about rings, ideals, nilpotent and Jacobson radicals, and the prime and Zariski spectra of rings.
(iii) The solutions cover topics such as when a sum of a nilpotent element and unit is a unit, when a polynomial is a unit or zero divisor based on its coefficients, and properties of the prime and Zariski topologies on the prime spectrum of a ring.
This document provides an overview and definitions related to functional analysis and Banach spaces. It discusses:
1) The definition of a Banach space as a complete normed linear space where every Cauchy sequence converges.
2) Examples of Banach spaces including l^p spaces, C(X) for compact X, C_b(X) for any topological space X, and C^k([a,b]).
3) Measure spaces and the definition of measurable functions on a measure space. It notes the closure properties of measurable functions under scalar multiplication and (sometimes) addition.
This document discusses the convexity of the set of k-admissible functions on a compact Kähler manifold. It begins by introducing k-admissible functions and some necessary convex analysis concepts. It then proves four main results: 1) the log of elementary symmetric functions of eigenvalues is a convex function, 2) the set of matrices with eigenvalues in a convex set is convex, 3) certain functions of eigenvalues are convex, and 4) the set of k-admissible functions is convex. It uses results on conjugation of spectral functions from convex analysis to prove these results. The proofs rely on properties of convex, lower semicontinuous functions and indicator functions.
The International Journal of Engineering & Science is aimed at providing a platform for researchers, engineers, scientists, or educators to publish their original research results, to exchange new ideas, to disseminate information in innovative designs, engineering experiences and technological skills. It is also the Journal's objective to promote engineering and technology education. All papers submitted to the Journal will be blind peer-reviewed. Only original articles will be published.
The papers for publication in The International Journal of Engineering& Science are selected through rigorous peer reviews to ensure originality, timeliness, relevance, and readability.
Theoretical work submitted to the Journal should be original in its motivation or modeling structure. Empirical analysis should be based on a theoretical framework and should be capable of replication. It is expected that all materials required for replication (including computer programs and data sets) should be available upon request to the authors.
Lecture13p.pdf.pdfThedeepness of freedom are threevalues.docxcroysierkathey
Lecture13p.pdf.pdf
Thedeepness of freedom are threevalues at thenude
functional Notconforming
patrtaf.us
vi sci x
I beease
ittouch
41 u VCsci inhalfedgeL
U VCI't x
Since u CPz are sci sc 7 that it have 3 Zeusunless e o
E is P anisolvent it forgiven I lop C P s t 4 p di
same
degree
y i l N Yi C E
Sabi n ofsystem YCp g
This is equivalent to say theonlypolynomial C PthetinterpolateZero
data Yifp o is the Zeno poly
vcpi.POTFF.gg In Edem
e e
I CRIvalue VCR Ca Ya
metfunctor
p
E3
pjJ Chip J
Shun E is p unisolvent
Y Cul VCR7 0
Xz V UCR o
rf VI UCB 0
Then over the edge PP we hone
C P havingtworootsPR
D This implies we 0
If e consider the other to edges G e b thesame
argument we can see Eo tht means W Lv o hersonly trivial
Solution
Then Yi CUI Ri for any Xi
E is P unisowent
y
csiy
Ya
f P Y cnn.PT
III Ldj Pg I Pre 2 ily
a PyO ein a
451214 7 f p i y g d CP f
ftp b f CRI I B so
fickle Cps O y Cp 7 L
Escaple5 in lectureto
Lecture_03_S08.pdf
LECTURE # 3:
ABSTRACT RITZ-GALERKIN METHOD
MATH610: NUMERICAL METHODS FOR PDES:
RAYTCHO LAZAROV
1. Variational Formulation
In the previous lecture we have introduced the following space of functions
defined on (0, 1):
(1)
V =
v :
v(x) is continuous function on (0, 1);
v′(x) exists in generalized sense and in L2(0, 1);
v(0) = v(1) = 0
:= H10 (0, 1)
and equipped it with the L2 and H1 norms
‖v‖ = (v,v)1/2 and ‖v‖V = (v,v)
1/2
V =
(∫ 1
0
(u′2 + u2)dx
)1
2
.
We also introduced the following variational and minimization problems:
(V ) find u ∈ V such that a(u,v) = L(v), ∀ v ∈ V,
(M) find u ∈ V such that F(u) ≤ F(v), ∀ v ∈ V,
where a(u,v) is a bilinear form that is symmetric, coercive and contin-
uous on V and L(v) is continuous on V and F(v) = 1
2
a(u,u) −L(v).
As an example we can take
a(u,v) ≡
∫ 1
0
(k(x)u′v′ + q(x)uv) dx and L(v) ≡
∫ 1
0
f(x)v dx.
Here we have assumed that there are positive constants k0, k1, M such that
(2) k1 ≥ k(x) ≥ k0 > 0, M ≥ q(x) ≥ 0, f ∈ L2(0, 1).
These are sufficient for the symmetry, coercivity and continuity of the
bilinear form a(., .) and the continuity of the linear form L(v).
The proof of these properties follows from the following theorem:
Theorem 1. Let u ∈ V ≡ H10 (0, 1). Then the following inequalities are
valid:
(3)
|u(x)|2 ≤ C1
∫ 1
0
(u′(x))2dx for any x ∈ (0, 1),∫ 1
0
u2(x)dx ≤ C0
∫ 1
0
(u′(x))2dx.
with constants C0 and C1 that are independent of u.
1
2 MATH610: NUMERICAL METHODS FOR PDES: RAYTCHO LAZAROV
Proof: We give two proofs. The simple one proves the above inequali-
ties with C0 = 1/2 and C1 = 1. The better proof establishes the above
inequalities with C0 = 1/6 and C1 = 1/4.
Indeed, for any x ∈ (0, 1) we have:
u(x) = u(0) +
∫ x
0
u′(s)ds.
Since u ∈ H10 (0, 1) then u(0) = 0. We square this equality and apply
Cauchy-Swartz inequality:
(4) |u(x)|2 =
∣∣∣∫ x
0
u′(s)ds
∣∣∣2 ≤ ∫ x
0
1ds
∫ x
0
(u′(s))2ds ≤ x
∫ x
0
(u′(s))2ds.
Taking the maximal value of x on the right hand side of this inequality
w ...
Lecture13p.pdf.pdfThedeepness of freedom are threevalues.docxjeremylockett77
The document discusses the abstract Ritz-Galerkin method for numerical solution of partial differential equations.
It begins by introducing the variational formulation and defining the function spaces used. It then describes the abstract form of the Ritz-Galerkin method, which finds an approximate solution in a finite dimensional subspace.
Specific examples are given to illustrate the method, including choice of basis functions for different boundary conditions. Basis functions for linear finite elements are defined on a partition of the domain into subintervals. The Ritz-Galerkin method results in a system of linear equations that can be solved for the coefficients of the approximate solution.
Remote Sensing and Computational, Evolutionary, Supercomputing, and Intellige...University of Maribor
Slides from talk:
Aleš Zamuda: Remote Sensing and Computational, Evolutionary, Supercomputing, and Intelligent Systems.
11th International Conference on Electrical, Electronics and Computer Engineering (IcETRAN), Niš, 3-6 June 2024
Inter-Society Networking Panel GRSS/MTT-S/CIS Panel Session: Promoting Connection and Cooperation
https://www.etran.rs/2024/en/home-english/
ESPP presentation to EU Waste Water Network, 4th June 2024 “EU policies driving nutrient removal and recycling
and the revised UWWTD (Urban Waste Water Treatment Directive)”
The ability to recreate computational results with minimal effort and actionable metrics provides a solid foundation for scientific research and software development. When people can replicate an analysis at the touch of a button using open-source software, open data, and methods to assess and compare proposals, it significantly eases verification of results, engagement with a diverse range of contributors, and progress. However, we have yet to fully achieve this; there are still many sociotechnical frictions.
Inspired by David Donoho's vision, this talk aims to revisit the three crucial pillars of frictionless reproducibility (data sharing, code sharing, and competitive challenges) with the perspective of deep software variability.
Our observation is that multiple layers — hardware, operating systems, third-party libraries, software versions, input data, compile-time options, and parameters — are subject to variability that exacerbates frictions but is also essential for achieving robust, generalizable results and fostering innovation. I will first review the literature, providing evidence of how the complex variability interactions across these layers affect qualitative and quantitative software properties, thereby complicating the reproduction and replication of scientific studies in various fields.
I will then present some software engineering and AI techniques that can support the strategic exploration of variability spaces. These include the use of abstractions and models (e.g., feature models), sampling strategies (e.g., uniform, random), cost-effective measurements (e.g., incremental build of software configurations), and dimensionality reduction methods (e.g., transfer learning, feature selection, software debloating).
I will finally argue that deep variability is both the problem and solution of frictionless reproducibility, calling the software science community to develop new methods and tools to manage variability and foster reproducibility in software systems.
Exposé invité Journées Nationales du GDR GPL 2024
What is greenhouse gasses and how many gasses are there to affect the Earth.moosaasad1975
What are greenhouse gasses how they affect the earth and its environment what is the future of the environment and earth how the weather and the climate effects.
The use of Nauplii and metanauplii artemia in aquaculture (brine shrimp).pptxMAGOTI ERNEST
Although Artemia has been known to man for centuries, its use as a food for the culture of larval organisms apparently began only in the 1930s, when several investigators found that it made an excellent food for newly hatched fish larvae (Litvinenko et al., 2023). As aquaculture developed in the 1960s and ‘70s, the use of Artemia also became more widespread, due both to its convenience and to its nutritional value for larval organisms (Arenas-Pardo et al., 2024). The fact that Artemia dormant cysts can be stored for long periods in cans, and then used as an off-the-shelf food requiring only 24 h of incubation makes them the most convenient, least labor-intensive, live food available for aquaculture (Sorgeloos & Roubach, 2021). The nutritional value of Artemia, especially for marine organisms, is not constant, but varies both geographically and temporally. During the last decade, however, both the causes of Artemia nutritional variability and methods to improve poorquality Artemia have been identified (Loufi et al., 2024).
Brine shrimp (Artemia spp.) are used in marine aquaculture worldwide. Annually, more than 2,000 metric tons of dry cysts are used for cultivation of fish, crustacean, and shellfish larva. Brine shrimp are important to aquaculture because newly hatched brine shrimp nauplii (larvae) provide a food source for many fish fry (Mozanzadeh et al., 2021). Culture and harvesting of brine shrimp eggs represents another aspect of the aquaculture industry. Nauplii and metanauplii of Artemia, commonly known as brine shrimp, play a crucial role in aquaculture due to their nutritional value and suitability as live feed for many aquatic species, particularly in larval stages (Sorgeloos & Roubach, 2021).
This presentation explores a brief idea about the structural and functional attributes of nucleotides, the structure and function of genetic materials along with the impact of UV rays and pH upon them.
Current Ms word generated power point presentation covers major details about the micronuclei test. It's significance and assays to conduct it. It is used to detect the micronuclei formation inside the cells of nearly every multicellular organism. It's formation takes place during chromosomal sepration at metaphase.
EWOCS-I: The catalog of X-ray sources in Westerlund 1 from the Extended Weste...Sérgio Sacani
Context. With a mass exceeding several 104 M⊙ and a rich and dense population of massive stars, supermassive young star clusters
represent the most massive star-forming environment that is dominated by the feedback from massive stars and gravitational interactions
among stars.
Aims. In this paper we present the Extended Westerlund 1 and 2 Open Clusters Survey (EWOCS) project, which aims to investigate
the influence of the starburst environment on the formation of stars and planets, and on the evolution of both low and high mass stars.
The primary targets of this project are Westerlund 1 and 2, the closest supermassive star clusters to the Sun.
Methods. The project is based primarily on recent observations conducted with the Chandra and JWST observatories. Specifically,
the Chandra survey of Westerlund 1 consists of 36 new ACIS-I observations, nearly co-pointed, for a total exposure time of 1 Msec.
Additionally, we included 8 archival Chandra/ACIS-S observations. This paper presents the resulting catalog of X-ray sources within
and around Westerlund 1. Sources were detected by combining various existing methods, and photon extraction and source validation
were carried out using the ACIS-Extract software.
Results. The EWOCS X-ray catalog comprises 5963 validated sources out of the 9420 initially provided to ACIS-Extract, reaching a
photon flux threshold of approximately 2 × 10−8 photons cm−2
s
−1
. The X-ray sources exhibit a highly concentrated spatial distribution,
with 1075 sources located within the central 1 arcmin. We have successfully detected X-ray emissions from 126 out of the 166 known
massive stars of the cluster, and we have collected over 71 000 photons from the magnetar CXO J164710.20-455217.
Nucleophilic Addition of carbonyl compounds.pptxSSR02
Nucleophilic addition is the most important reaction of carbonyls. Not just aldehydes and ketones, but also carboxylic acid derivatives in general.
Carbonyls undergo addition reactions with a large range of nucleophiles.
Comparing the relative basicity of the nucleophile and the product is extremely helpful in determining how reversible the addition reaction is. Reactions with Grignards and hydrides are irreversible. Reactions with weak bases like halides and carboxylates generally don’t happen.
Electronic effects (inductive effects, electron donation) have a large impact on reactivity.
Large groups adjacent to the carbonyl will slow the rate of reaction.
Neutral nucleophiles can also add to carbonyls, although their additions are generally slower and more reversible. Acid catalysis is sometimes employed to increase the rate of addition.
Phenomics assisted breeding in crop improvementIshaGoswami9
As the population is increasing and will reach about 9 billion upto 2050. Also due to climate change, it is difficult to meet the food requirement of such a large population. Facing the challenges presented by resource shortages, climate
change, and increasing global population, crop yield and quality need to be improved in a sustainable way over the coming decades. Genetic improvement by breeding is the best way to increase crop productivity. With the rapid progression of functional
genomics, an increasing number of crop genomes have been sequenced and dozens of genes influencing key agronomic traits have been identified. However, current genome sequence information has not been adequately exploited for understanding
the complex characteristics of multiple gene, owing to a lack of crop phenotypic data. Efficient, automatic, and accurate technologies and platforms that can capture phenotypic data that can
be linked to genomics information for crop improvement at all growth stages have become as important as genotyping. Thus,
high-throughput phenotyping has become the major bottleneck restricting crop breeding. Plant phenomics has been defined as the high-throughput, accurate acquisition and analysis of multi-dimensional phenotypes
during crop growing stages at the organism level, including the cell, tissue, organ, individual plant, plot, and field levels. With the rapid development of novel sensors, imaging technology,
and analysis methods, numerous infrastructure platforms have been developed for phenotyping.
aziz sancar nobel prize winner: from mardin to nobel
Let XandY be two sets.docx
1. Let X and Y be two sets; then the Cartesian product Z = X x Y is
the set of all ordered pairs ( x , y) with x E X and y E Y. We shall first
show that the Cartesian productof two measurable spaces ( X , X)
and ( Y, Y) can be made into a measurable space in a natural fashion.
Next we shall show that if measures are given on each of the factor
spaces, we can define a measure on the productspace. Finally, we
shall relatejntegration with respectto the productmeasure and iterated
integration with respect to the measures in the factor spaces. The
model to be kept in mind throughout this discussion is the plane, which
we regard as the productR x R.
10.1 DEFINITION. If (X, X) and ( Y, Y) are measurable spaces, then a
set of the form A x B with A E X and B E Y is called a measurable
rectangle, or simply a rectangle, in Z = X x Y, We shall denote the
collection of all finite unions of rectangles by Zo.
It is an exercise to show that every set in Zo can be expressed as a
finite disjoint union of rectangles in Z (see Exercise 10.D).
10.2 LEMMA. Thecollection Zo is an algebra of subsets of Z .
PROOF. It is clear that the union of a finite number of sets in Zo
also belongs to Zo. Similarly, it follows from the first part of Exercise
10.E that the complement of a rectangle in Z belongs to Zo. Apply
De Morgan's laws to see that the complement of any set in Zo belongs
to Zo. Q.E.D.
11 3
1 14 The Elements of Integration
10.3 DEFINITION. If (X, X) and ( Y, Y) are measurable spaces, then
Z = X x Y denotes the a-algebra of subsets of Z = X x Y generated
by rectangles A x B with A E X and B E Y. We shall refer to a set in
Z as a Z-measurable set, or as a measurable subset of Z.
If (X, X, p) and (Y, Y, v) are measure spaces, it is natural to attempt
to define a measure r on the subsets of Z = X x Y which is the
"product" of p and v in the sense that
(Recall the convention that O(fco) = 0.) We shall now show that
this can always be done.
10.4 PRODUCTM EASURTEH EOREM.I f (X, X, p) and (Y, Y, v) are
measure spaces, then there exists a measure r defined on Z = X x Y
such that
for all A E X and B E Y. If these measure spaces are a-finite, then there
is a unique measure r with property (10.1).
PROOF. Supposethat the rectangle A x B is the disjoint union of a
2. sequence (A, x Bj) of rectangles; thus
for all x E X, y E Y. Hold x fixed, integrate with respect to v, and
apply the Monotone Convergence Theorem to obtain
A further application of the Monotone Convergence Theorem yields
Now let E E ZO; without loss of generality we may assume that
Product Measures 1 15
where the sets A, x B, are mutually disjoint rectangles. If we define
no(E) by
the argument in the previous paragraph implies that no is well-defined
and countably additive on 2,. By Theorem 9.7, there is an extension
of 7r0 to a measure n on the a-algebra Z generated by Zo. Since n is
an extension of no, it is clear that (10.1) holds.
If (X, X, p) and (Y, Y, v) are a-finite, then no is a 5-finite measure
on the algebra Zo and the uniqueness of a measure satisfying (10.1)
follows from the uniqueness assertion of the Hahn Extension Theorem
9.8. Q.E.D.
Theorem 10.4 establishes the existence of a measure n on the U-algebra
Z generated by the rectangles (A x B : A E X, B E Y) and such that
(10.1) holds. Any such measure will be called a productof p and v.
If p and v are both a-finite, then they have a unique product. In the
general casethe extension procedurediscussed in the previous chapter
leads to a uniquely determined productmeasure. However, it will be
seen in Exercise 10.S that it is possible for two distinct measures on Z
to satisfy (10.1) if p and v are not a-finite.
In order to relate integration with respect to a productmeasure and
iterated integration, the notion of a section is useful.
10.5 DEFINITION. If E is a subset of Z = X x Y and x E X, then
the x-section of E is the set
Ex = {YE Y : (x, y) E E)
Similarly, if y E Y, then the y-section of E is the set
Iff is a function defined on Z to ii, and x E X, then the x-section off
is the functionf, defined on Y by
The Elements of Integration
Similarly, if y E Y, then the y-section off is the function f Y defined
on X by
f y ( x ) = f ( x , ~ ) , XEX.
10.6 LEMMA. (a) If E is a measurable subsetof 2, then every section
of E is measurable.
(b) Iff is a measurable function on Z to 8, then every section off is
measurable.
3. PROOF. (a) If E = A x B and x E X, then the x-section of E is B
if x E A, and is 0 if x # A. Therefore, the rectangles are contained in
the collection E of sets in Z having the property that each x-section is
measurable. Since it is easily seen that E is a U-algebra (see Exercise
10.I), it follows that E = 2 .
(b) Let x E Xand a E R, then
If f is 2-measurable, then f, is Y-measurable. Similarly, f Y is Xmeasurable.
Q.E.D.
We interpolate an important result, which is often useful in measure
and probability theory, and which will be used below. We recall (see
Exercise 2.V) that a monotone class is a nonempty collection M of sets
which contains the union of each increasing sequence in M and the
intersection of each decreasing sequence in M. It is easy (see Exercise
2.W) to show that if A is a nonempty collection of subsets of a set S,
then the U-algebra S generated by A contains the monotone class M
generated by A. We now show that if A is an algebra, then S = M.
10.7 MONOTONCE LASSLEMMA.1 f.A is an algebra of sets, then the
a-algebra Sgenerated by A coincides with the monotone class M generated
by A.
PROOF. We have remarked that M G S. To obtain the opposite
inclusion it suffices to prove that M is an algebra.
If EE M, define M(E) to be the collection of FE M such that
E F, E n F, F E all belong to M. Evidently 0, E E M(E) and it is
Product Measures 1 17
readily seen that M(E) is a monotone class. Moreover, F E M(E) if
and only if E E M(F).
If E belongs to the algebra A, then it is clear that A s M(E).
But since M is the smallest monotone class containing A, we must have
M(E) = MforEinA. T h e r e f o r e , i f E ~ A a n d F ~ M , t h e n F ~ M ( E ) .
We infer that if E E A and F E M, then E E M(F) so that A s M(F) for
any FE M. Using the minimality of M once more we conclude that
M(F) = M for any FE M. Thus M is closed under intersections and
relative complements. But since X E M it is plain that M is an algebra ;
since it is a monotone class, it is indeed a a-algebra. Q.E.D.
It follows from the Monotone Class Lemma that if a monotone class
contains an algebra A, then it contains the a-algebra generated by A.
10.8 LEMMA. Let (X, X, ,u) and ( Y, Y, v) be a-finite measure spaces.
If E E Z = X x Y, then the functions defined by
are measurable, and
4. J fdp = n(E) = J gdv.
X Y
PROOF. First we shall supposethat the measure spaces are finite
and let M be the collection of all E E Z for which the above assertion
is true. We shall show that M = Z by demonstrating that M is a
monotone class containing the algebra Zo. In fact, if E = A x B
with A E X and B E Y, then
Since an arbitrary element of Zo can be written as a finite disjoint
union of rectangles, it follows that Z, G M.
We now show that M is a monotone class. Indeed, let (En) be a
monotone increasing sequence in M with union E, Therefore
1 1 8 The Elements of Integration
are measurable and
Jt is clear that the monotone increasing sequences (A) and (gn) converge
to the functions f and g defined by
If we apply the fact that n is a measure and the Monotone Convergence
Theorem, we obtain
so that EE M. Since n is finite measure, it can be proved in the same
way that if (F,) is a monotone decreasing sequence in M7 then F = n F,
belongs to M. Therefore M is a monotone class, and it follows from
the Monotone Class Lemma that M = 2.
If the measure spaces are a-finite, let Z be the increasing union of a
sequence of rectangles (Z,) with n(Zn) < +a and apply the previous
argument and the Monotone Convergence Theorem to the sequence
(E n Zn) Q.ED. .
10.9 TONELLI'ST HEOREM.L et (X, X, ,u) and ( Y, Y, v) be a-finite
measure spaces and let F be a nonnegative measurable function on
Z = X x Y to R. Then the functions defined on X and Y by
are measurable and
(1 0.5) IX fd,u = J' Fdn = IYgdv.
z
In other symbols,
PROOF. If F is the characteristic function of a set in 2 , the assertion
follows from the Lemma 10.8. By linearity, the present theorem holds
Product Measures 1 19
for a measurable simple function. If F is an arbitrary nonnegative
5. measurable function on Z to k, Lemma 2.11 implies that there is a
sequence (0,) of nonnegative measurable simple functions which
converges in a monotone increasing fashion on Z to F. If yn and +, are
defined by
then yn and +,, are measurable and monotone in n. By the Monotone
Convergence Theorem, (y,) converges on X to f and (i,hn) converges on
Y to g. Another application of the Monotone Convergence Theorem
implies that
f dp = lim yn dp = lim IZ 0. d~
X
The same theorem also shows that
from which (10.5) follows. Q.E.D.
It will be seen in the exercises that Tonelli's Theorem may fail if we
drop the hypothesis that F is nonnegative, or if we drop the hypothesis
that the measures p, v are a-finite.
Tonelli's Theorem deals with a nonnegative function on Z and
affirms the equality of the integral over Z and the two iterated integrals
whether these integrals are finite or equal +a. The final result
considers the casewhere the function is allowed to take both positive
and negative values, but is assumed to be integrable.
10.10 FUBINI'TS HEOREM.L et (X, X, ,u) and ( Y, Y, v) be a-Jinite
spaces and let the measure T on Z = X x Y be the product of p and v. I f
the function F on Z = X x Y to R is integrable with respect to T, then
the extended real-valued functions deJned almost everywhere by
120 The Elements of Integration
have finite integrals and
In other symbols,
PROOF. Since F is integrable with respectto n, its positive and
negative parts F + and F- are integrable. Apply Tonelli's Theorem
to F+ and F- to deduce that the carresponding f + and f- have finite
integrals with respectto p. Hence f+ and f- are finite-valued
p-almost everywhere, so their difference f is defined p-almost everywhere
and the first part of (10.9) is clear. The second part is similar.
Q.E.D.
Since we have chosenin Chapter 5 to restrict the use of the word
"integrable" to real-valued functions, we cannot conclude that the
6. functions f, g defined in (10.8) are integrable. However, they are
almost everywhere equal to integrable functions.
It will be seen in an exercise that Fubini's Theorem may fail if the
hypothesis that F is integrable is dropped.
EXERCISES
10.A. Let A s X and B s Y. If A or B is empty, then A x B = 0.
Conversely, if A x B = 0, then either A = 0 or B = 0.
10.B. Let A, s Xand Bj s Y, j = 1,2. If Al x B1 = A2 x B2
# 0, then Al =A2 and B1 = B2.
10.C. Let A, s Xand Bj s Y, j = 1,2. Then
and the sets on the right side are mutually disjoint.
Product Measures 121
10.D. Let (X, X) and (Y, Y) be measurable spaces. If Aj EX and
Bj E Y for j = 1,. . . , m, then the set
can be written as the disjoint union of a finite number of rectangles in Z.
10.E. Let Aj G X and B, s Y, j = 1,2. Then
10.F. If (R, B) denotes the measurable space consisting of real
numbers together with the Borel sets, show that every open subsetof
R x R belongs to B x B. In fact, this 5-algebra is the 5-algebra
generated by the open subsets of R x R. (In other words, B x B is
the Borel algebra of R x R .)
10.G. Let f and g be real-valued functions on X and Y, respectively;
supposethat f is X-measurable and that g is Y-measurable. If h is
defined for (x, y) in X x Y by h(x, y) = f(x)g(y), show that h is
X x Y-measurable.
10.H. IfEis asubset ofR, let y(E) = {(x, y )R~ x R : x - y E E).
If E E B, show that y(E) E B x B. Use this to prove that if ,f is a
Borel measurable function on R to R, then the function F defined by
F(x, y) = f(x - y) is measurable with respect to B x B.
10.1. Let E and F be subsets of Z = X x Y, and let x E X. Show
that (E E), = Ex Fx. If (Ea) are subsets of Z, then
10.J. Let (X, X, tc) be the measure space on the natural numbers
X = N with the counting measure defined on all subsets of X = N.
Let (Y, Y, v) be an arbitrary measure space. Show that a set E in
Z = X x Y belongs to Z = X x Y if and only if each section En of E
belongs to Y. In this casethere is a unique productmeasure T, and
122 The Elements of Integration
A function f on Z = X x Y to R is measurable if and only if each
7. section f, is Y-measurable. Moreover, f is integrable with respectto T
if and only if the series
is convergent, in which case
10.K. Let X and Y be the unit interval [0, 11 and let X and Y be the
Bore1 subsets of [0, 11. Let p be Lebesgue measure on X and let v be
the counting measure on Y. If D = {(x, y) : x = y}, show that D is a
measurable subsetof Z = X x Y, but that
Hence Lemma 10.8 may fail unless both of the factors are required to
be a-finite.
10.L. If F is the characteristic function of the set D in the Exercise
10.K, show that Tonelli's Theorem may fail unless both of the factors
are required to be a-finite.
10.M. Show that the example considered in Exercise 10.J demonstrates
that Tonelli's Theorem holds for arbitrary (Y, Y, v) when
(X, X, p) is the set N of natural numbers with the counting measure on
arbitrary subsets of N.
10.N. If am, 2 0 for m, n E N, then
10.0. Let a,, be defined for m, n E N by requiring that a,, = + 1,
a,,,,, = -1,anda,, = Oifm # n o r m # n + 1 . Showthat
so the hypothesis of integrability in Fubini's Theorem cannot be
dropped,
Product Measures 1 23
1O.P. Let f be integrable on (X, X, ,u) , let g be integrable on ( Y, Y, Y) ,
and define h on Z by h(x, y) = f(x) g(y). If n is a productof p and Y,
show that h is n-integrable and
10.Q. Supposethat (X, X, ,u) and (Y, Y, Y) are a-finite, and let
E, F belong to X x Y. If v(E,) = v(F,) for all x E X, then n(E) = n(F).
10.R. Let f and g be Lebesgue integrable functions on (R, B) to R.
From Exercise 10.H it follows that the function mapping (x, y) into
f(x - y)g(y) is measurable with respectto B x B. If X denotes
Lebesgue measure on B, use Tonelli's Theorem and the fact that
to show that the function h defined for x E R by
is finite almost everywhere. Moreover,
The function h defined above is called the convolution off and g and is
usually denoted by f * g.
10.S. Let X = R, X be the a-algebra of all subsets of R and let p be
defined by p(A) = 0 if A is countable, and p(A) = +co if A is uncountable.
We shall constructdistinct products of p with itself.
8. (a) If E E Z = X x X, define n(E) = 0 in case E can be written as
the union E = G u H of two sets in Z such that the x-projection of G
is countable and the y-projection of H is countable. Otherwise, define
n(E) = +a. It is evident that n is a measure on Z. If n(E) = 0,
then E is contained in the union of a countable set of lines in the plane.
If A, B E X, show that n(A x B) = p(A) p(B). Hence n is a product
of p with itself.
124 The Elements of Integration
(b) If EE 2 , define p(E) = 0 in case E can be written as the union
E = G u H u K of three sets in Z such that the x-projection of G is
countable, the y-projection of H is countable, and the projection of K
on the line with equation y = x is countable. Otherwise, define
p(E) = +co. Now p is a measure on 2, and if p(E) = 0, then E is
contained in the union of a countable set of lines. Show that
p(A x B) = p(A) p(B) for all A, B E X; hence p is a productof p with
itself.
(c) Let E = {(x, y) : x + y = 0); show that E E Z . However,
p(E) = 0, whereas r(E) = + co .