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The subsequent statement can be regarded as a follow-up to

Let $f:[a,b]\to\mathbb{R}$ be Lebesgue integrable. Furthemore, let $$ g:[a,b]\ni x\mapsto\int_a^x f(t)\,\mathrm{d}t\in\mathbb{R} $$ be convex. Then $f$ is non-decreasing almost everywhere.

Let $a\le x_0<x_1<x_2\le b$. Since $f$ is convex, we have $$ \frac{g(x_2)-g(x_1)}{x_2-x_1}-\frac{g(x_1)-g(x_0)}{x_1-x_0}\ge 0\text{.} $$ This can be reduced to $$ \int_{x_1}^{x_2} \frac{f(t)}{x_2-x_1}\,\mathrm{d}t \ge\int_{x_0}^{x_1} \frac{f(t)}{x_1-x_0}\,\mathrm{d}t\text{.} $$ The last formula roughly shows that the 'average' $f$ on $[x_0,x_1]$ does not exceed the 'average' of $f$ on $[x_1,x_2]$. Do you know a rigorous argument showing that $f$ is non-decreasing a.e.?

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precarious
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    What does "non-decreasing almost everywhere" even mean? – TonyK Jan 20 '12 at 21:02
  • @TonyK It means that there exists a set $X\subseteq [a,b]$ of measure $b-a$ such that for all $x_0,x_1\in X$ with $x_0\le x_1$ it holds $f(x_0)\le f(x_1)$. – precarious Jan 20 '12 at 21:10

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A standard result about convex functions is that they have nondecreasing one-sided derivatives everywhere. Since $g'(x)=f(x)$ almost everywhere, $f$ is equal almost everywhere to the nondecreasing right-hand derivative of $g$.

Jonas Meyer
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  • Thank you very much. This is the kind of argument I was looking for. Can you recommend literature where I can find a deep treatment of convex functions? – precarious Jan 21 '12 at 11:07
  • @precarious: Try "Convex functions: constructions, characterizations and counterexamples" by Jonathan M. Borwein, Jon D. Vanderwerff. http://carma.newcastle.edu.au/ConvexFunctions/CF-CCC.pdf – matt Jan 21 '12 at 11:45
  • @matt Thank you for your recommendation. I had a quick look at the table of contents and the first chapter. I am not sure whether this is what I am looking for. The book seems to cover quite general settings while I am focused on univariate functions given on finite intervals. Maybe, a survey article about this topic would fit my needs best. – precarious Jan 21 '12 at 12:32
  • @precarious: I don't know if it is deep, but there is a good summary (with proofs) of some basic properties of convex functions in one variable in section 7.6 of Wheeden and Zygmund's Measure and integral, starting on page 118. – Jonas Meyer Jan 21 '12 at 19:22
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    @Jonas: Thank you. This reference is similar to what I was looking for. In the meantime, I also found Convex Functions by Roberts and Varberg. Maybe 'deep' was said too much. On the other hand, these results are striking even if they seem to be folklore. – precarious Jan 21 '12 at 20:43