Differentiation of a series of increasing functions - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T19:37:53Z http://mathoverflow.net/feeds/question/51139 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/51139/differentiation-of-a-series-of-increasing-functions Differentiation of a series of increasing functions Analyst2 2011-01-04T17:18:50Z 2011-01-05T00:02:18Z <p>Hi,</p> <p>Let $(f_n)_{n \geq 1}$ be a sequence of increasing functions defined on an interval, say $[0,1]$. </p> <p>Suppose that $\sum_{n=1}^{\infty}f_n(x)$ converges for all $x \in [0,1]$. Let $f:=\sum_{n=1}^{\infty}f_n$. </p> <p>It is well known that an increasing function defined on an interval is differentiable almost everywhere on that interval. But is it true that</p> <p>$$f'(x)= \sum_{n=1}^{\infty}f_n'(x)$$ almost everywhere on $[0,1]$?</p> <p>Any reference would help.</p> <p>Thank you.</p> http://mathoverflow.net/questions/51139/differentiation-of-a-series-of-increasing-functions/51143#51143 Answer by Shai Covo for Differentiation of a series of increasing functions Shai Covo 2011-01-04T17:51:16Z 2011-01-04T18:24:05Z <p>Yes, see Theorem 4.1 on p. 177 of <a href="http://books.google.co.il/books?id=5ddbKSkaL8EC&amp;pg=PR8&amp;lpg=PR8&amp;dq=%2522differentiating+series+of+monotone+functions%2522&amp;source=bl&amp;ots=_L6toIMGqu&amp;sig=oTTRArrbAIjIKSFBRs6Egb819Ts&amp;hl=iw&amp;ei=IWYjTbK0BYGyhAfOnby3Dg&amp;sa=X&amp;oi=book_result&amp;ct=result&amp;resnum=1&amp;ved=0CAYQ6AEwAA#v=onepage&amp;q=%2522differentiating%2520series%2520of%2520monotone%2520functions%2522&amp;f=false" rel="nofollow">this book</a>.</p> http://mathoverflow.net/questions/51139/differentiation-of-a-series-of-increasing-functions/51164#51164 Answer by Byron Schmuland for Differentiation of a series of increasing functions Byron Schmuland 2011-01-05T00:02:18Z 2011-01-05T00:02:18Z <p>This is also Theorem 17.18 (page 267) of <em>Real and Abstract Analysis</em> by Hewitt and Ross. The result is credited there to Fubini.</p>