Exercise 1.6.14 (Locally integrable functions)

We demonstrate that both definitions of the local absolute integrability are indeed equivalent.

• Suppose that for any $x\in {ℝ}^d$ we can find an open neighbourhood of $x$ on which the integral of $f$ is finite. The trick is to notice how this condition implies that $f$ is absolutely integrable on every compact set in $R^n$. To see why, let $K\subset {ℝ}^d$ be an arbitrary compact set. For each $x\in K$ we can find an open set $O_x$ on which $f$ is absolutely integrable. But obviously, ${\bigcup <!--\; FUNCTION\; APPLICATION\; -->}_{x\in K}{O}_k$ is an open cover for $K$, and so we can find a finite subcover $O_1,\dots ,O_n$ from ${(O_x)}_{x\in K}$ such than ${\bigcup <!--\; FUNCTION\; APPLICATION\; -->}_{i=1}^n{O}_i$ covers $K$. By additivity of the Lebesgue integral, we then obtain

  $${\int \; <!--\; nolimits\; -->}_K\left|f\right|dm\le \sum _{i=1}^n{\int \; <!--\; nolimits\; -->}_{O_i}\left|f\right|dm<\infty .$$

  Now consider the ball $B(0,r)$ for arbitrary radius $r>0$. The boundary of this ball $\mathit{\partial B}(0,r)$ is a null set, and so

  $${\int \; <!--\; nolimits\; -->}_{B(0,r)}\left|f\right|dm={\int \; <!--\; nolimits\; -->}_{\overline{B(0,r)}}\left|f\right|dm-0<\infty$$

  since $\overline{B(0,r)}$ is compact by Heine-Borel theorem.

• Suppose that ${\int }_{B(0,r)}\left|f\right|<\infty$ for any $r>0$. Pick an arbitrary $x\in {ℝ}^d$. We cannot use $B(0,r)$ since it is not centred at $x$. The trick is to look at ${\int }_{B(0,2\parallel x\parallel )}\left|f\right|<\infty$ on $B(0,2\parallel x\parallel )$ instead. It is easy to verify using the triangle inequality that $B(x,\parallel x\parallel )\subseteq B(0,2\parallel x\parallel )$, and so we have found a neighbourhood $B(x,\parallel x\parallel )$ of $x$ at which it is guaranteed that ${\int }_{B(x,\parallel x\parallel )}\left|f\right|<\infty$. Since our choice of $x\in {ℝ}^d$ was arbitrary, the claim follows.

This result can be used to generalise Theorem 1.6.19 (Lebesgue differentiation theorem in general dimension) to any locally integrable function $f$. This is done by considering a point $x\in {ℝ}^d$, finding a open neighbourhood $U$ of ${ℝ}^d$ on which $f$ is absolutely integrable. We then apply Theorem 1.6.19 to the restriction $f\upharpoonright U$ (one would of course choose $r>0$ small enough so that the balls $B(0,r)$ in the limit pass inside the $U$).