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.

1. Measure Theory Cheat Sheet
A collection T os subsets of a set X is said to be a topology in X if T satisfies
the following properties,
• ∅ ∈ T and X ∈ T
• Closed under finite intersections
• Closed under arbitrary unions
Members of T are called open sets.
If X, Y are topological spaces then f : X → Y is continuous if f−1
(V ) is open
in X for all open sets V ∈ Y .
Topology
A collection F of subsets of X is called a σ−algebra if the following properties
hold
• X ∈ F
• If A ∈ F then AC
= X − A ∈ F
• Closed under unions
σ-algebra
• If F is a σ−algebra of X then X is a measurable space and members of
F are measurable sets in X.
• If X is a measurable space and Y is a topological space, then f : X → Y
is said to be measurable if f−1
(V ) is a measurable set in X for all open
sets V in Y .
Characteristic function: It is a measurable function defined as follows .If E is
a measurable set in X define χE(x) =
(
1 if x ∈ E
0 if x 6∈ E
Measureability
Generated σ−algebra: For any collection of subsets F of X there exists a
smallest σ−algebra which contains F. It is the intersection of all σ−algebras
containing F. Denote it as σ(F).
Borel σ− algebra: For a topological space X the σ−algebra generated by the
family of open sets of X. Elements of a Borel σ−algebra are called Borel sets.
Borel mapping: A map between two topological spaces f : X → Y if the
inverse image of an open set in Y is an element of the Borel σ−algebra of X.
• If f : X → [−∞, ∞] and F is a σ−algebra of X, then f is measurable if
f−1
((a, ∞)) ∈ F for all a.
Borel σ−algebra
• If fn : X → [−∞, ∞] is measurable for all n ∈ N then sup, inf, lim sup,
lim inf of fn are also measurable.
• The limit of every pointwise convergent sequence of measurable func-
tions is measurable.
• If f is measurable then so is f+
= max{f, 0} and , f−1
= − min{f, 0}
Pointwise convergence and measurability
A complex function whose range consists of only finitely many points. If
α1 . . . , αn are the distinct values of the simple function s and Ai = {x : s(x) =
αi} then
s =
n
X
i=1
αiχAi
• Every measurable function f : X → [0, ∞] can be written as a pointwise
limit of a sequence of simple functions.
Simple functions
A positive measure µ is a measure along with the following additional prop-
erties,
• Its range is in [0, ∞]
• Countable additivity: µ(
S∞
i=1 Ai) =
P∞
i=1 µ(Ai)
A measure space refers to a measurable space with a positive measure.
Positive measure
We understand a + ∞ = ∞ for 0 ≤ a ≤ ∞ and a · ∞ = ∞ if 0 < a ≤ ∞ else 0.
Arithmetic in [0, ∞]
If X is a set with σ−algebra F and positive measure µ. Then for a measurable
simple function s : X → [0, ∞] as defined previously, its Lebesgue integral
over E ∈ F is defined as follows
ˆ
E
s dµ =
n
X
i=1
αiµ
Ai
E
Lebesgue integral
Define L1
(µ) to be the collection of all complex measurable functions f on X
for which
´
X
|f| dµ ∞ known as the Lebesgue integrable functions.
For functions f with range in [−∞, ∞] define
´
E
f dµ =
´
E
f+
dµ −
´
E
f−
dµ
Lebesgue integrable functions
We say a property holds “almost everywhere (a.e.)” if it holds everywhere
except on a set of measure zero.
If any two function f = g a.e. then their Lebesgue integrals are the same. Set
of measure zero don’t impact the value of the Lebesgue integral
Zero measure
Let {fn} be a sequence of measurable functions on X if 0 ≤ f1(x) ≤ f2(x) ≤
· · · ≤ ∞ a.e. and fn(x) → f(x) as n → ∞ a.e., then f is measurable and
lim
n→∞
ˆ
X
fn dµ =
ˆ
X
lim
n→∞
fn dµ
Monotone convergence theorem
• Applying MCT to sequence of partial sums of a convergent series
f(x) =
P∞
n=1 fn(x) where fn : X → [0, ∞] measurable for all n we
get,
ˆ
X
f dµ =
∞
X
n=1
ˆ
X
fn dµ
• If f : X → [0, ∞] is measurable with σ−algebra F of X and φ(E) =
´
E
f dµ for E ⊂ F then, φ is a measure on F and
ˆ
X
g dφ =
ˆ
X
gf dµ
for every measurable g : X → [0, ∞].
Consequences of MCT
If fn : X → [0, ∞] is measurable for all n ∈ N then,
ˆ
X
lim inf
n→∞
fn dµ ≤ lim inf
n→∞
ˆ
X
fn dµ
Fatou’s lemma
If {fn} us a sequence of complex measurable functions on X with
limn→∞ fn(x) = f(x) pointwise. If there exists a function g ∈ L1
(µ) such
that |fn(x)| ≤ g(x) for all n ∈ N and x ∈ X then f ∈ L(
1)(µ) and
lim
n→∞
ˆ
X
f dµ =
ˆ
X
lim
n→∞
fn(x) dµ
Dominated convergence theorem
A measure is called complete if all subsets of sets of measure 0 are measurable.
Every measure can be completed.
Complete measure
A function has compact support if it is zero outside of a compact set, i.e.
f ∈ Cc(X) if f has compact support on X.
Compact support
A topological space X is called locally compact if every point x ∈ X has a
compact neighbourhood.
Uryshon’s lemma: If X is a locally compact Hausdorff space. Let K ⊆ X is
compact and U open s.t. K ⊆ U ⊂ X, there exists f ∈ Cc(X) with 0 ≤ f ≤ 1
such that fK ≡ 1 and f ≡ 0 otherwise.
Locally compact Hausdorff spaces
Let X is a locally compact Hausdorff space and T is a positive linear func-
tional on Cc(X). Then there exists a σ−algebra F of X which contains all
Borel sets in X and there exists a unique positive measure µ on F such that
for every f ∈ Cc(X)
Tf =
ˆ
X
f dµ
additionally the following properties hold,
1. µ(K) ∞ for every compact K ⊂ X
2. µ(E) = inf{µ(V ) : E ⊂ V, V open} for all E ∈ F.
3. µ(E) = sup{µ(K) : K ⊂ E, Kcompact} holds for every open set E ⊂ F
with µ(E) ∞
4. If E ∈ F, A ⊂ E and µ(E) = 0 then A ∈ F.
Riesz representation theorem
A set in a measure space is said to have a σ−finite measure if it is the countable
union of sets of finite measure.
A set in a topological space is said to be σ−compact if it is the countable union
of compact sets.
σ−finite measure
A measure µ defined on the σ−algebra of all Borel sets in a locally compact
Hausdorff space X.
If µ is positive we also say a Borel set is,
• Outer regular if satisfies property 3 of the above theorem.
• Inner regular if it satisfies property 4 of the above theorem.
• Regular if it is both inner and outer regular.
In a locally compact σ−compact Hausdorff space X. If there exists a positive
Borel measure µ defined on X such that for K ⊆ X compact µ(K) ∞ =⇒
µ is regular.
Regular Borel measures

2. A family of subsets S of a set X is called a semiring of sets if it satisfies the
following properties,
• ∅ ∈ S
• A, B ∈ S =⇒ A
T
B ∈ S
• There exists finite Ki ∈ S pairwise disjoint s.t. A B =
Sn
i=1 Ki
Semiring of sets
A function µ from a semiring of sets S to [0, ∞] is called a premeasure if
• µ(∅) = 0
• µ(
S∞
i=1 Ai) =
P∞
i=1 µ(Ai) and
S∞
i=1 Ai ∈ S
Premeasure
For a set X, semiring S of X and a pre-measure µ : S → [0, ∞] there exists a
unique extension to a measure µ̃ : σ(A) → [0, ∞].
Carathéodory extension theorem
For the semiring S = {[a, b) : a, b ∈ R, a ≤ b} say [a, b) = I with
µ([a, b)) = b−a = `(I) we have (by the previous theorem) a unique extension
µ̃ on the Borel sets of R this measure is the Lebesgue measure.
Existence and Uniqueness of Lebesgue measure
The Lebesgue outer measure for an arbitrary subset E ⊆ R is defined as
µ∗
(A) = inf{
P∞
i=1 `(Ik)} over a countable family of open intervals {Ik} that
cover A.
Lebesgue outer measure
A ⊆ R is Lebesgue measurable iff µ∗
(B) = µ∗
(B
T
A) + µ∗
(B A) for all
B ∈ R.
Carathéodory’s criterion
The Lebesgue measure µ for R is defined on the sigma algebra of the sets
satisfying Carathéodory’s criterion as µ(A) = µ∗
(A).
• Every Borel set is Lebesgue measurable, but the converse need not be
true.
• µ is translation invariant, µ(A + x) = µ(A) for every Lebesgue measur-
able A and every x ∈ R
Lebesgue measure
Consider µ1 a measure defined on σ−algebra F1 of X1 and µ2 measure de-
fined on σ-algebra F2 of X2. Then the product measure µ̃ = µ1 × µ2 is the
Carathéodory extension of the premeasure µ(A × B) = µ(A) · µ(B).
• Note that the product σ−algebra is the generated σ−algebra of F1 × F2
as the ordinary Cartesian product is only a semi ring.
• The product measure is unique if µ1, µ2 are σ−finite.
Product measures
Similarly as defined above for σ−finite measure spaces
ˆ
X×Y
f(x, y) d(µ1 × µ2) =
ˆ
Y
ˆ
X
f(x, y) dµ1
dµ2
=
ˆ
X
ˆ
Y
f(x, y) dµ2
dµ1
Fubinis theorem
A real function ϕ : [a, b] → R is convex if ϕ((1−λ)x+λy) ≤ (1−λ)ϕ(x)+λϕ(y)
for x, y ∈ (a, b) and λ ∈ [0, 1].
Convexity implies continuity.
Convexity
Let µ is a positive measure on a σ−algebra F in a set X such that µ(X) = 1. If
f is a real function in L1
(µ), if a f(x) b for all x ∈ X and if ϕ convex on
(a,b), then
ϕ
ˆ
X
f dµ ≤
ˆ
X
(ϕ ◦ f) dµ
Jensen’s inequality
Let p, q ∈ R s.t. 1
p + 1
q = 1, 1 ≤ p ≤ ∞, and measure µ on X. If f, g : X → [0, ∞]
measurable then,
ˆ
X
fg dµ ≤
ˆ
X
fp
dµ
1
p
ˆ
X
gq
dµ
1
q
Hölder’s inequality
Let p, q ∈ R s.t. 1
p + 1
q = 1, 1 ≤ p ≤ ∞, and measure µ on X. If f, g : X → [0, ∞]
measurable then,
ˆ
X
(f + g)p
dµ
1
p
≤
ˆ
X
fp
dµ
1
p
ˆ
X
gq
dµ
1
q
Minkowski’s inequality
For X with positive measure µ if 0 p ∞ if f is complex measurable on X
define the Lp
norm as
||f||p =
ˆ
X
|f|p
dµ
1
p
and let Lp
(µ) be the collection of all functions for which Lp
norm is finite. If
th measure is the counting measure we denote it as `p
.
Define ||f||∞ = sup{|f(x)| : x ∈ X}
• Lp
is a complete metric space.
• Cc(X) is dense in Lp
(µ).
Lp
norms
For any complex measure µ we define its total variation |µ| defined for some
measurable set E as
|µ|(E) = sup
( ∞
X
i=1
|µ(Ei)|
)
where supremum is taken over all of partitions of measurable subsets Ei of
E.
• Total variation is a positive measure.
• The total variation of a positive measure is the same as the positive
measure itself.
• Total variation of any measurable set is always finite.
Total variation
An arbitrary measure λ is absolutely continuous with respect to a positive
measure µ, denoted as λ µ if λ(E) = 0 for every measurable E for which
µ(E) = E.
λ is said to be concentrated on measurable set A if λ(A) = λ (A
T
E) for every
measurable set E.
For two measures λ1, λ2 if there exist disjoint measurable sets A, B such that
λ1 concentrated on A and λ2 concentrated on B ten we say the measures are
mutually singular, i.e. λ1⊥λ2.
Absolutely continuity
Let µ be a positive σ−finite measure on a σ−algebra F on set X and let λ be
a complex measure on F. Then,
• There exists a unique pair of complex measures λa, λs on F such that
λ = λa + λs and λa µ, λs⊥µ
• There exists a unique h ∈ L1
(µ) such that
λa(E) =
ˆ
E
h dµ
for every E ∈ F.
The integral defines a absolutely continuous measure wrt µ and h is referred
to as the Radon-Nikodym derivative of λa wrt µ.
Lebesgue-Radon-Nikodym theorem
• Alternate statement for absolute continuity: If for every ε 0 there ex-
ists a δ 0 such that µ(E) δ =⇒ |λ(E)| ε for every measurable E,
then λ µ.
• Polar decomposition of λ : For complex measure λ there exists a measur-
able function h such that |h(x)| = 1 for all x ∈ X such that dµ = hd|µ|
• Hahn decomposition theorem: For real measure µ on σ− algebra in set
X then there exists sets A, B in F such that A
S
B = X, A
T
B = ∅ and
µ+
(E) = µ(A ∩ E), µ−
(E) = −µ(B ∩ E) for measurable E.
Consequences of Radon-Nikodym Theorem