Let $(f_n)_{n \geq 1}$ be a sequence of increasing functions defined on an interval, say $[0,1]$.

Suppose that $\sum_{n=1}^{\infty}f_n(x)$ converges for all $x \in [0,1]$. Let $f:=\sum_{n=1}^{\infty}f_n$.

It is well known that an increasing function defined on an interval is differentiable almost everywhere on that interval. But is it true that

$$f'(x)= \sum_{n=1}^{\infty}f_n'(x)$$ almost everywhere on $[0,1]$?

Any reference would help.

Thank you.


Yes, see Theorem 4.1 on p. 177 of this book.


This is also Theorem 17.18 (page 267) of Real and Abstract Analysis by Hewitt and Ross. The result is credited there to Fubini.


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