Lecture 19 - Feb 17

Suppose , where is integrable on .

Define

As

Therefore

Then, and is injective.

Hence by DCT,

Assume

Exercise: prove this.

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Theorem

Let be Riemann integrable where with . Then is Lebesgue integrable on .

Then

Proof

Given a partition

Define

NOTE

If is a refinement of , then

Prove this.

Lemma

if and only if there exists a sequence of partitions

such that

1. is a refinement of

2. Length of , as

Suppose

Define

Observe

Also,

NOTE

That and

and are called monotonic sequences in .

By applying Fatou’s Lemma to

NOTE

{Also show measurability}

But

Note that , where

Hence by DCT,

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