Show that $f$ is measurable on $(a,b)$

Question

Let $f,F : (a,b)\to \mathbb{R}$ such that $F$ is differentiable on $(a,b)$ and $F'(x) =f(x)$ for each $x \in (a,b)$. Show that $f$ is measurable on $(a,b)$. Hint : Show first that $\forall x \in (a,b) : f(x) = \lim\limits_{n \to \infty} n[F(x+\frac{1}{n}) - F(x)]$.

I have already proved the hint ... How to proceed please?

score 0 · Accepted Answer · answered Nov 23 '18 at 22:04

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Note that each function $x\mapsto n(F(x + 1/n) - F(x))$ is measurable and that the limit of a sequence of measurable functions is measurable.

answered Nov 23 '18 at 22:04

ncmathsadist

49,383

We can say this because : Since F is differentiable then it is continuous and so it is measurable. Isn't it? – Ahmed Nov 23 '18 at 22:14
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That is correct. – ncmathsadist Nov 23 '18 at 22:17

Show that $f$ is measurable on $(a,b)$

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