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Let $f \colon \mathbb{R} \to \mathbb{R}$ be an increasing function. Prove or disprove: $f$ is Lebesgue measurable.

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marked as duplicate by Jonas Meyer, Cameron Buie, Danny Cheuk, JSchlather, Calvin Lin Sep 24 '13 at 5:12

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Yes. It's a standard result that $f$ is measurable iff $f^{-1}([\alpha, \infty))$ is measurable for each $\alpha$, and such a pre-image must be either empty or an interval when $f$ is monotone.

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oh, it was very easy xD why didn't I see that!? thank you! – user67133 Sep 24 '13 at 2:59