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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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    Some related questions: http://math.stackexchange.com/questions/252421/are-monotone-functions-borel-measurable?rq=1, http://math.stackexchange.com/q/105094/, http://math.stackexchange.com/questions/229497/f-mathbfr-rightarrow-mathbfr-monotone-increasing-rightarrow-f-is. – Jonas Meyer Sep 24 '13 at 03:14

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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.