# Lebesgue-Stieltjes Integration

Question

Let $$\alpha:[a,b]\rightarrow\mathbb{R}$$ be a function with finite total variation.

Let $$F:\mathbb{R}\rightarrow\mathbb{R}$$ be differentiable on [a,b].

Denote $$\Delta\alpha(s):=\alpha(s)-\alpha(s-)$$ where $$\alpha(s-)$$ is the left limit in s for $$s>a$$ and $$\alpha(a-)=\alpha(a)$$. Show that

$$F(\alpha(x))=F(\alpha(a))+\int^{x}_{0}F(\alpha(s-))d\alpha(s)+\sum_{s\leq t}[F(\alpha(s))-F(\alpha(s-))-F'(\alpha(s-))\Delta\alpha(s )]$$

Help! Any hints on where to start? And where/why does the sum come from?