What's the reason for the pre-image def. of Lebesgue measurable?
I.e. $f$ Lebesgue measurable if $\{ x : f(x) > c \in \mathbb{R}\}$ is measurable.
Like why is it significant that $f(x) > c$?
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1An equivalent intuitive characterization is that measurable functions are the ones that can be approximated by simple functions. – Qi Zhu Nov 12 '19 at 14:07
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The gist here is that it implies immediately that we can measure $f$ on intervals which is the key idea for Lebesgue integrals. – Qi Zhu Nov 12 '19 at 14:08