Measure Theory — Part 2

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In part 1, I have introduced the concept of σ-algebra Σ, measure, measurable set, measurable function, simple function and Lebesgue Integral.

Let’s continue our journey on the measure theory…

σ-Algebra Generator

Recall the defintion of a measurable function, if we want to prove that a function f: X → Y is measurable, we need to prove the pre-image of all measurable sets B in Σy is in Σx. However, with the help of the σ-algebra Generator, we can prove this in a simpler way when Σy satisfies certain condition.

SA4. Intersection of σ-algebra is σ-algebra
Proof
Let Σ1 and Σ2 be σ-algebra on set X.
SA1: X ∈ Σ1 and X ∈ Σ2
⇒ X ∈ Σ1∩Σ2
SA2:
∀A ∈ Σ1∩Σ2 ⇒ A ∈ Σ1 and A ∈ Σ2
⇒ X\A ∈ Σ1 and X\A ∈ Σ2 ⇒ X\A ∈ Σ1∩Σ2
SA3:
for all countable collection A[i] ∈ Σ1∩Σ2,
⇒ A[i] ∈ Σ1 and A[i] ∈ Σ2
⇒ A[1]∪…∪A[∞] ∈ Σ1 and A[1]∪…∪A[∞] ∈ Σ2
⇒ A[1]∪…∪A[∞] ∈ Σ1∩Σ2
⇒ Σ1∩Σ2 is σ-algebra

Let C be a (countable or uncountable) collection of subsets of the set X. Let G(C) be the σ-algebra…