This is my first PPT made for a college assignment on Measure Theory and Advanced Probability. I tried to cover the topic "Counting measure" under combinatorics from all dimensions according to my humble knowledge.

1. Counting Measures
1
Measure Theory and Advanced Probability
By Saloni Singhal

2. Basic Theory
Counting measures plays a fundamental role
in discrete probability structures, and
particularly those that involve sampling from a
finite set. The sample space set S is typically
large, hence efficient counting measures are
essential.
It is an intuitive way to put a ‘measure’ on any
set. Counting measure can be defined on any
measurable space (any set X along with a
sigma-algebra).

3. 0 1 2
This is a finite countable set whose measure
is defined in terms of cardinality which
Is equal to the number of elements,
here n=3
This is an interval on the number line
[0,1]. Since it has many elements hence its
Measure is infinity.
Let {X,B) be a measurable space, the measure μ on X defined by
μ(A) =. n, if n has exactly n elements
∞, otherwise

4. Counting measure is simply
summation!
The function # on P(S) its called counting
measure. In many cases set of objects can be
counted by establishing a one-to-one
correspondence between them.
Addition rule: If {A1,A2…} are collection of
disjoint sets then

5. Bool’s inequality
Bonferroni’s inequality
Inclusion -Exclusion formula
Generalization:

6. The multiplication rule of combinatorics is based on the formulation of a
procedure (or algorithm) that generates the objects to be counted.
key to a successful application of the multiplication rule to a counting problem
is the clear formulation of an algorithm that generates the objects being
counted, so that each object is generated once
Product Sets

7. Terminology
L(I) is the length of interval I
O=UIi,(i=1 to infinity), length of open set
m(E) measure of E (extended notion of
length fnc)
m*(E) outer measure =inf Σl(I)

8. Measurable Set
E is said to be measurable if for each set A
we can define, m*(A)=m*(A∩E)+M*(A∩Ec),
i.e. these sets (bounded or unbounded)split
set into two pieces (measurable or non-
measurable that are additive wrt outer
measure.
Countable union of such sets is also
measurable.
Every Borel set is measurable.

9. Properties
m(E)≥0
m(ϕ)=0
Monotonicity: A⊂b, m(B)≥m(A)
Countably additive:m(⋃Ei)=Σm(Ei)
Translational invariance; m(A+x)=m(A)
m(E)= 0 if E is countable and 1 if Ec is countable

10. Non-Measurable Sets
Most of the sets we come across in analysis are
measurable.
Several example of non-measurable set were given
by G. Vitali(1905), Van Vleck(1908)
Robert Solvey(1970) proved existence of non
measurable set can’t be established if axiom of
choice is disallowed.
Every set of positive measure contains a non-
measurable set.

11. Integration with Counting Measures
Let μ be counting measure on a set Ω. (This measure is not σ-finite unless Ω is
countable.)
If A ⊆ Ω, then μ(A) = #(A), the number of elements in A. If f is a nonnegative simple
function, f =
􏰈
ΣaiIAi , (i=1,,,n), then
Measurable Functions
Let f be an extended real-valued/unction defined on a measurable set E (of finite or
infinite measure). Then the following statements are equivalent:
􏰋n

12. References
en.wikipedia.org/wiki/Counting_measure
www.biblio.com/book/lebesgue-measure-
integration-p-k-jain/d/741911691
stats.libretexts.org/Bookshelves/Probability_The
ory/Book3A_Probability_Mathematical_Statistics
_and_Stochastic_Processes_
www.stat.cmu.edu/~arinaldo/Teaching/36752/S1
8/Notes/lec_notes_2.pdf

13. –Saloni Singhal
“I thank Dr. Qazi Azhad Jamal Sir for his
‘immeasurable’ support, guidance
and words of wisdom throughout
this journey.”