Differentiation of a series of increasing functions

Question

Hi,

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.

Answer 1

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

Answer 2

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