Proof of Lebesgue Differentation Theorem in Lipschitz Analysis Lectures by Heinonen

I was reading Lectures on Lipschitz Analysis by Juha Heinonen and at the Theorem 3.2 he gives a proof of Lebesgue Differentiation Theorem for monotone functions. He says that we can easily check the following:

• $q|E_q| \leq |f(E_q)|$,
• $|f(E_p)| \leq p|E_p|$.

Firstful it doesn't seem to be defined in the Lecture notes what $E_{q}$ and $E_{p}$ are. And also I don't how one can easily come to this conclusion. Most proofs of this theorem that I found are much more lengthy. Could anyone shed some light on this?

Solution 1:

Part 1. This is the answer to your question "What is the definition of $E_q$?".

The author's statement after equation (3.6): "if D^+f(x) > q at every x \in E_q"

could be written more clearly as this:

Definition. The set $E_q=\{ x: D^{+} f(x)>q\}$.

Intuitively, this set $E_q$ consists of all points $x$ for which there exists "arbitrarily small" open intervals $I$ that

(i) contain $x$ and satisfy

(ii) $\frac{ |f(I)|}{|I|} >q$.

Note: The number of such $I$ can be uncountable, and they overlap one another.

Part 2. This answers your request to have the details of the Vitali Lemma application explained intuitively.

The goal is to use properties (i) and (ii) to deduce the measure-theoretic inequality

(iii) $\frac{|f(E_q)|}{ |E_q|}\geq q$.

Arguing naively, if we could somehow show that $E_q$ can be expressed as a countable disjoint union of intervals $I_k$ that each satisfy (ii), then by summing over all such terms $I_k$ in that disjoint decomposition we would expect to obtain (iii).

This naive argument unfortunately cannot be literally correct: $E$ itself is not always expressible as a countable disjoint union of intervals $I_k$ that satisfy (ii). Some technical tricks are needed.

First approximate $E$ from the outside by any reasonably well-chosen open set $G\supset E$ so that we can get a good outer-measure approximation to $E$. Consider the open intervals $I$ that satisfy (i), (ii), and also $I\subset G$.

The Vitali Covering Lemma is precisely the tool required to show that one can extract a countable DISJOINT subcollection of open intervals $I_k$ satisfying all of the above and that cover almost all of $E$, (missing only a subset of $E$ that has measure zero).

The proof of the Vitali theorem can be found many places and is too lengthy to discuss here. It is the quintessential prototypical example of a Greedy Algorithm that consists of a countable number of steps, in which one picks the largest possible $I$ at each stage.