StudyLabs

Lecture 33 - Apr 13

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Recap and Motivation

Let .

If is differentiable and is Riemann integrable, then

The converse is false in general.

Typical example:

Here is differentiable, but is not Riemann integrable on .

Also recall the Cantor function :

  1. , .
  2. is continuous and increasing.
  3. a.e.

So conditions stronger than a.e. differentiability are needed for a Newton-Leibniz type formula.

Definition (Absolute Continuity)

A function is called absolutely continuous (A.C.) if:

For every , there exists such that for any finite collection of pairwise disjoint open intervals

with

we have

Remark

Every absolutely continuous function is uniformly continuous, but uniform continuity does not imply absolute continuity.

Proposition

Absolute continuity implies bounded variation:

Proof idea from class

Fix for in the A.C. definition. Given any partition , group its subintervals into at most

blocks, each of total length . On each block, the sum of increments is . Hence the total variation along is bounded by , uniformly in . Therefore .

Lemma (Absolute Continuity of Integral)

Let be a measure space and . Then given , there exists such that

Proof sketch

If a.e., choose . In general, truncate by

use monotone convergence to choose with

then combine with bounded case for .

Proposition

Let and define

Then is absolutely continuous on .

Proof

Fix and let come from the lemma for . If are pairwise disjoint intervals with

then for ,

So .

Main Theorem (Characterization of Absolute Continuity)

For ,

Equivalently, for absolutely continuous , one may take

and recover

Class note: this direction uses the Radon-Nikodym theorem in the standard measure-theoretic proof.

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