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I was having trouble with what I believe to be a rather easy problem from section six of Terence Tao's Introduction to measure theory. Help will be appreciated.

If $F$ is everywhere differentiable (and thus $F$ is continuous), show $F'$ is measurable.

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1 Answer

Define:

$$ g_n(x) = \frac{F(x + 1/n) - F(x)}{1/n} $$

This is a sequence of measurable functions. The pointwise limit as $n \to \infty$ is $F'$. Since the pointwise limit of a sequence of measurable functions is measurable, it follows that $F'$ is measurable.

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