Mesure theory: First

Table of content

• Set
• Index set
• Mesure theory
• Power set
• Sigma algebra

Set

Set is a collection of different things.

Index Set

Index Set: The index set I is any set such that for i ∈ I, we refer to A_i, which is also any set.

Example: If I = {1, 2, 3}, and A_1 = {1, 3, 5}, A_2 = {2, 4}, A_3 = {∅}.

Measure Theory

Measure theory is a generalization and formalization of:

• Geometrical measures like length, volume, area
• Other notations like magnitude, mass, probability of events

Example: Giving a set [a, b] on R, the length is b - a

Power Set

Power set P(X) of a set X is a set of all subsets of X.

Example: If X = {1, 2}, then P(X) = {∅, X, {1}, {2}}.

Proving the Size of the Power Set

If X has n elements, then P(X) has 2ⁿ elements. Let's prove it:

• If n = 0, then X = ∅. This is true for P(X) = {∅}, which has 1 = 2⁰ element.
• Let n > 0 and k > 0.
• Suppose n = k, then the statement is correct. Now, we need to prove it for n = k + 1.
• Let X have n = k + 1 elements, or X = {x₁, x₂, ... , x_k+1}.
• The power set P(X) is composed of:
  • Subsets that don't contain x_k+1 (denoted as P₁)
  • Subsets that do contain x_k+1 (denoted as P₂)
• By assumption, when n = k, P₁ has 2^k elements.
• P₂ can be formed by taking each element of P₁ and adding x_k+1 to it, so P₂ also has 2^k elements.
• Therefore, P(X) = P₁ + P₂ has 2^k + 2^k = 2^k+1 elements.

Sigma Algebra of a Set

A sigma algebra of a set X, denoted F, must satisfy the following:

• X ∈ F
• Closed under complementation: Let A ∈ X, then X \ A = A^c ∈ F
• Closed under countable unions: Let A_i ∈ F with i ∈ N, then a finite or infinite union of A_i ∈ F

Example: X = {a, b, c}, list all sigma algebras of X:

• F = {∅, X}
• F = {∅, X, {a}, {b, c}}
• F = {∅, X, {b}, {a, c}}
• F = {∅, X, {c}, {a, b}}
• F = {∅, X, {a, b}, {c}}
• F = {∅, X, {a, c}, {b}}
• F = {∅, X, {b, c}, {a}}
• F = {∅, X, {a}, {b}, {c}, {a, b}, {a, c}, {b, c}}

Example: Let X = {a, b, c} and F = {∅, X, {a}, {b, c}}. Let's analyze why we can't insert {a, b} into F:

Consider inserting {a, b} to make:

F = {∅, X, {a}, {b, c}, {a, b}}

But X \ {a, b} = {c} ∉ F → F is no longer a sigma algebra.

Now consider inserting {c} to make:

F = {∅, X, {a}, {b, c}, {a, b}, {c}}

But a finite union {a} ∪ {c} = {a, c} doesn't belong to F → F is no longer a sigma algebra.

Proof that the Countable Intersection of Sigma Algebras is a Sigma Algebra

Let F_i be sigma algebras of X with i ∈ I and I is an index set.
Let F = ⋂ F_i with i ∈ I.
We need to prove that F is a sigma algebra.

1. Contains X and ∅

Since every F_i is a sigma algebra of X, then X and ∅ belong to each F_i.
By definition of intersection, F = ⋂ F_i also contains X and ∅.

2. Closed under Complements

Let a set A belong to F.
This means A also belongs to every F_i.
By definition of a sigma algebra, A^c = X \ A is also in every F_i.
By definition of intersection, A^c is also in ⋂ F_i = F.

3. Closed under Countable Unions

Let A₁, A₂, ..., A_n be a countable set of sets inside F.
Then A₁, A₂, ..., A_n are also in every F_i.
By definition of a sigma algebra, ⋃ A_i with i=1, n is also in each F_i.
By definition of intersection, ⋃ A_i with i=1, n is also in ⋂ F_i = F.

Conclusion: F is also a sigma algebra of X.