Theorem
Suppose are integrable with
Then pointwise almost everywhere.
Proof
Theorem
are Lebesgue measurable and .
Then for any , is also measurable and .
Hint:
Proof
As , is also measurable (by completeness).
Proposition
, . Suppose for any we have or . Then every set in also has the same property.
Proof
Clearly, then we need to show is a -algebra.
Proposition
such that , .
Take
Proposition
Let
Definition
is differentiable everywhere, is differentiable everywhere.
Given measurable function.
Define .
Case 1: Fix
Case 2:
If , then then converges to .
If , then .
Take
Then is Borel measurable &
Theorem
is a non-decreasing sequence of non-negative measurable functions. Suppose then
Proof
By Fatou’s Lemma,
Given ;