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Mesurable Functions by definition(from Stein and Shakarchi):

A function $f$ defined on a measurable subset $E$ of $\mathbb{R}^d$ is measurable, if for all $a\in \mathbb{R}$, the set $$f^{-1}([-\infty,a))=\{x\in E: f(x) is measurable.

Now a set $E$ is called measurable if $m_*(E)=0.$

Intuitively, the definition doesn't make much sense to me and would appreciate it if someone can explain it to me. A bonus would be if you can give me some simple examples of measurable and non measurable functions? Thanks.

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    That is not the definition of a set being measurable.2012-03-27
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    @ChrisEagle Probably the sigma algebra contains all sets of measure zero in his definition so all these sets are measurable though this is still not a valid definition of measurable sets2015-06-09

3 Answers 3