I have a monotone real function $f$ defined on [0,1] with values in [0,1]. The function need not be continuous. I need to make sense of the expression: $F(a)=\int_0^a f'(x)x dx,$ in the greater possible generality and in the most elementary way. What is the best way to go about it?
How to make sense of $\int_0^a f'(x)x dx$ for monotone, not necessarily continuous, functions $f$
2
$\begingroup$
real-analysis
integration
-
0@GilKalai: Maybe you should interpret the integral as the [Riemann-Stieltjes](http://en.wikipedia.org/wiki/Riemann%E2%80%93Stieltjes_integral) integral $\int_0^ax\mathrm{d}f(x)$? This seems quite general to me. – 2012-05-05
1 Answers
5
Let us pretend for a moment that $f$ is regular, then an integration by parts yields $ F(a)=\left[xf(x)\right]_{0}^a-\int_0^af(x)\mathrm dx=af(a)-\int_0^af(x)\mathrm dx. $ Since the RHS is meaningful for every monotonic function $f$ (and still others), one can define $F(a)$ by the expression in the RHS for every such function.
-
0Thanks, Gil. (Nice blog, by the way... :-)) – 2012-05-05