Lecture 31 - Apr 10

Continuation: Lebesgue-Young Theorem

Theorem

If

is monotonically increasing, then is differentiable a.e. (w.r.t. Lebesgue measure).

From previous lecture, it is enough to show that the exceptional sets are null:

where

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Step 1: Show

Fix and .

For each , there exists such that

Hence intervals

form a Vitali cover of .

By Vitali covering lemma, choose pairwise disjoint intervals

such that

Then

But by construction,

so

Therefore

Let and then , obtaining

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Step 2: Show (same for )

For rational numbers , define

Then

So it is enough to prove

Fix and choose open such that

From , for each choose with

Thus is a Vitali cover of .

By Vitali, choose disjoint intervals

with

Set

Then

Now use for : choose right intervals

which form a Vitali cover of . Again by Vitali, choose disjoint

such that

Hence

Also,

From the first family,

Comparing the two estimates through monotonicity/telescoping on disjoint intervals inside gives

Letting and using forces

Therefore . Similarly, .

Combining Steps 1 and 2, is differentiable a.e. on .

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Remark

The proof above is the standard Vitali-covering argument for monotone functions and is the key step in showing monotone functions are a.e. differentiable.