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I'm studing Evans PDE book, especifically, the proof of theorem 6 here.

Let $U\subset\mathbb{R}^n$ be an open set and $f:U\to\mathbb{R}$ a locally integrable function. In the proof is taken an open set $V\subset\mathbb{R}^n$ such that $B\subset V\subset\overline{V}\subset U$ (where $B$ is a closed ball that contains the support of $\eta_\varepsilon$).

As $f$ is locally integrable, we can conclude that $f$ is integrable over $\overline{V}$. However I need $f$ integrable over $V$. My analysis book says that if $X,Y\subset\mathbb{R}^n$ are both Jordan measurable and $f:X\cup Y\to\mathbb{R}$ is integrable then $f$ is integrable over $X$ and over $Y$.

So, my question is: can we ensure that $V$ and $\partial V$ are both Jordan measurable? Or, to be more precise, given an open set $U\subset\mathbb{R}^n$ and a closd ball $B\subset U$, is there an open set $V\subset U$ such that (i) $V\supset B$ and (ii) $V$ and $\partial V$ are both Jordan Measurable? (if so, I think we can conclude that $f$ is integrable over $V$ because it's integrable over $\overline{V}=V\cup\partial V$).

Thanks.

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Pedro
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Evans is not concerned with Jordan measurability. It seems that your analysis book does not provide adequate background for reading Evans. You should find a book that does Lebesgue integration. (Real Analysis by Folland is one of such.) Every Borel set is Lebesgue measurable.

user98130
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  • So, can we use the property 7 here to conclude that $f$ is (Lebesgue) integrable over $V$ because $V$ and $\partial V$ are both borel sets? – Pedro Oct 07 '13 at 12:00
  • @Pedro Yes, although the fact you need is something that's more basic than that. Namely: if $f:X\to\mathbb R$ is integrable, and $A$ is a measurable subset of $X$, then the restriction of $f$ to $A$ is integrable. – user98130 Oct 07 '13 at 12:04
  • Do you know if is there any context in which mollifiers are defined with Riemann integrals? – Pedro Oct 07 '13 at 12:12
  • @Pedro I don't. I don't know anyone who uses Riemann integrals or Jordan measures to do analysis outside of undergraduate textbooks. – user98130 Oct 07 '13 at 12:23