How to make sense of $\int_0^a f'(x)x dx$ for monotone, not necessarily continuous, functions $f$ 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?

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.
