# Approximating bounded measurable sets by compact sets from the inside: reference needed

Does anyone know a precise reference on the following assertion? (The question has already been discussed here.)

Let $m$ denote the Lebesgue measure, and let $A\subset \mathbb{R}^d$ be $m$-measurable and bounded. Then, for every $\varepsilon>0$ there is a compact set $K_{\varepsilon}\subset A$ such that $m(A\setminus K_{\varepsilon})<\varepsilon$.

Many thanks in advance! Peter

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See baby Rudin Chapter 10 statement 10.11 –  Norbert May 14 '12 at 9:34
This property of the Lebesgue measure is called inner-Regularity. –  Alex Becker May 14 '12 at 9:45
@Norbert I think you mean chapter 11? (this is the case for the third edition at least). –  user22705 May 14 '12 at 10:56
I use Russian edition, this the reason of difference. –  Norbert May 14 '12 at 11:01
Thanks a lot. That's just the kind of reference I've been searching for. –  Peter May 14 '12 at 12:36