Note. This note is based on the seventh chapter of Zygmund's well-known textbook, Measure and integral. Some proofs will be skipped or left only ideas. You may find the proofs in Zygmund's book. Please tell me if there is any mistake in my note :).
0. Foreword
In the first part of this note, we will derive the relationship between Lebesgue integral and differentiation, which is the Lebesgue differentiation theorem. In the proof of this theorem, we introduce the simple Vitali lemma, which will be refined in the next. Then we will use this refined lemma to prove the differentiation of a monotone increasing function exists almost everywhere, and this fact is immediately generalized to function of bounded variation. Also, in the same theorem, we note that
\[0\leq \int_{a}^{b} f' \leq f(b-)-f(a+).\]
The last inequality cannot be replaced by equality, even if $f$ is continuous on $[a,b]$ (Consider the Cantor-Lebesgue function). To deal with this question, we introduce the concepts of absolute continuity and Singularity, and then we know that the equality holds if we added that $f$ is absolutely continuous. On the other hand, we will find out that a function of BV can split into a absolutely continuous function plus a singular function. Also, we will prove the Lebesgue version of integration by parts.
I skipped the part of convex function in the last part of chapter 7 in Zygmund's book.
