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In this question, I am specifically interested in the following case:

Let f:R->R be a Borel-measurable function from the reals to the reals and let

$g(x)\ =\ \int_{b}^x\ f(t)\ dt$,

where the integral is the Lebesgue integral and b is a real constant. I would presume

$g'(x) = f(x) $ a. e.

Is this true? Is there a published reference for this? (It might be an exercise in a book.) I tried looking in Rao's encyclopedic compendium on Lebesgue measure and integration, among others, but couldn't seem to find an answer.

smbmath
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  • If I remember correctly, the riemann and lebesgue integral are the same on riemann integrable functions. You would only have to check for $L^1$-limits of Riemann integrable functions. – Vincent Batens Dec 24 '23 at 02:29
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    Look up Lebesgue differentiation theorem – fwd Dec 24 '23 at 02:51

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