Exercise 1.6.17 (Lebesgue differentiation theorem II)

We have all of the ingredients to use the dense subclass trick.

• Theorem proven for a dense subclass
  In Exercise 1.6.16 we have shown that the result holds for any continuous function, i.e., for any continuous function $g:{ℝ}^d\to ℂ$ and almost any $x\in {ℝ}^d$ we have

  $$\underset{r\to 0}{\lim <!--\; FUNCTION\; APPLICATION\; -->}\frac{1}{m(B(x,r))}{\int \; <!--\; nolimits\; -->}_{B(x,r)}|g-g(x)|dm=0.$$

• Quantititaive estimate on how close the a function and its approximation from the dense subclass can get
  Let $𝜖>0$ and $\lambda >0$ be arbitrary. By Littlewood’s second principle, we can find a compactly supported continuous function $g:{ℝ}^d\to ℂ$ such that

  $${\int \; <!--\; nolimits\; -->}_{{ℝ}^d}\left|f-g\right|dm\le 𝜖$$

  Applying the Hardy-Littlewood inequality, we obtain

  $$m\left(\left\{x\in {ℝ}^d:\underset{r>0}{\sup <!--\; FUNCTION\; APPLICATION\; -->}\frac{1}{m(B(x,r))}{\int \; <!--\; nolimits\; -->}_{B(x,r)}|f-g|dm\ge \lambda \right\}\right)\le \frac{C_d}{\lambda }{\int \; <!--\; nolimits\; -->}_{{ℝ}^d}|f-g|dm\le C_d\frac{𝜖}{\lambda }$$

  In a similar spirit, Markov’s inequality tells us

  $$m\left(\left\{x\in {ℝ}^d:{\int \; <!--\; nolimits\; -->}_{{ℝ}^d}|f-g|dm\ge \lambda \right\}\right)\le \frac{1}{\lambda }{\int \; <!--\; nolimits\; -->}_{{ℝ}^d}|f-g|dm\le \frac{𝜖}{\lambda }$$

  Thus, let $E$ be the exceptional set

  $$E:=\left\{x\in {ℝ}^d:\underset{r>0}{\sup <!--\; FUNCTION\; APPLICATION\; -->}\frac{1}{m(B(x,r))}{\int \; <!--\; nolimits\; -->}_{B(x,r)}|f-g|dm\ge \lambda \right\}\cup \left\{x\in {ℝ}^d:{\int \; <!--\; nolimits\; -->}_{{ℝ}^d}|f-g|dm\ge \lambda \right\}$$

  By subadditivity of Lebesgue measure, the set $E$ has a measure of at most $(C_d+1)\cdot 𝜖∕\lambda$. We then obtain two quantitative estimates:

  (i)

  for all $x\in {ℝ}^d∕E$: $|f(x)-g(x)|\le \lambda$.

  (ii)

  for all $x\in {ℝ}^d∕E$: $1∕m(B(x,r)){\int }_{B(x,r)}|f-g|\mathit{dm}<\lambda$ for all $r>0$.

We now put both of these ingredients together. Let $g$ be the continuous function from the quantitative estimate part. From the dense subclass result we can find a $r>0$ small enough so that

We have by triangle inequality for almost every $x\in {ℝ}^d∕E$ with $m(E)\le (C_d+1)\cdot 𝜖∕\lambda$:

for all $r>0$ sufficiently close to zero. In particular, this implies

Keeping $\lambda >0$ fixed, and sending $𝜖>0$ to zero (by taking infimum w.r.t. $𝜖$), we conclude that

Taking limits as $\lambda \to 0$ afterwards, we conclude by non-negativity of the integral that

and the claim follows.