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When I learned about the Lebesgue integral, the definition I was taught of measurable set was the following: $A\subseteq\mathbb{R}^n$ is measurable if and only if $1_A$ is an almost everywhere limit of step functions.

However, these days I saw on the Internet another definition of measurability: $A$ is measurable if and only if for all $B\subseteq\mathbb{R}^n$ one has $m(A\cap B)+m(A^c\cap B)=m(B)$, where $m$ is the Lebesgue measure.

I would like to know the relation between these two definitions.

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  • $\begingroup$ Well given $(X, \Omega)$ a measurable space, $A \subset X$ is measurable if $A \in \Omega$. That's the sole definition of a measurable set in modern approach to the subject. $\endgroup$ – Hermès Oct 3 '16 at 19:31
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    $\begingroup$ I'm actually baffled that you were first taught this definition of a measurable set. $\endgroup$ – Hubble Oct 3 '16 at 19:34
  • $\begingroup$ @Hermès And which is the set $\Omega$ in the case of $\mathbb{R}^n$? That is, what is a measurable set? Does $\Omega$ correspond to any of the definitions I posted? $\endgroup$ – hie Oct 3 '16 at 19:34
  • $\begingroup$ @user372906, $\Omega$ is a $\sigma$-algebra on $X$. $\endgroup$ – Hubble Oct 3 '16 at 19:35
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    $\begingroup$ @Hubble My teacher followed the procedure of the book "Mathematical Analysis", by Apostol. First, the Lebesgue integral is described with no measure theory (only the concept of null set), and then the concept of measurable function is defined: $f$ is measurable iff $f$ is the almost everywhere limit of step functions. A set $A$ is measurable iff $1_A$ is measurable. In this way, from the Lebesgue integral, you construct measure theory. $\endgroup$ – hie Oct 3 '16 at 19:37

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