Excision property for non measurable sets

Asked Jul 08 '19 at 09:24

Active Jul 08 '19 at 09:33

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I'm studying measure theory from the book Real Analysis by H. Royden & P. Fitzpatrick. I'm stuck at the question 19 of the chapter 2.

The question is:

Let $E$ has finite outer measure. If $E$ is not measurable, then there is an open set $O$ containing $E$ that has finite measure and for which $m^*(O\setminus E) > m^*(O) - m^*(E)$

Applying the definition of measurability, i get a set satisfying the inequality but cannot find an open set.

PS: I've found a duplicate but it has no answers. So i decided to ask again.

edited Jul 08 '19 at 09:33

asked Jul 08 '19 at 09:24

user550494

Try regularity of Lebesgue measure? – Jakobian Jul 08 '19 at 09:27
@bof just fixed my mistake. It should be outer measure. – user550494 Jul 08 '19 at 09:34
@Jakobian i have tried but it gives me only an $\epsilon $ with $m^*(O\setminus E) > \epsilon $ for all open sets $O$ containing $E$. How can i continue to get RHS of the inequality? – user550494 Jul 08 '19 at 09:41
It boils down to showing that Catheodory criterion of measurability doesn't hold for every such open set $O$. – Jakobian Jul 08 '19 at 09:52

Excision property for non measurable sets

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