Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 202 / 1 / b / Solution

Refine a dyadic partition by inserting $s$ and $t$. The triangle inequality shows that its variation over $[s,t]$ is at least $|f(t)-f(s)|$. Passing to the defining limit givesConsequently the càdlàg functionsare nondecreasing: for $s\le t$, the displayed inequality makes both increments nonnegative. Thus they are distribution functions in the Stieltjes sense and $f=F-G$; this is the Jordan decomposition of a function of bounded variation.

Now $A_f\subseteq A_F\cup A_G$, so $A_f$ is countable by part (a). For any finite subset $J\subseteq A_f\cap (0,t]$, partitions isolating its points and the triangle inequality giveTaking the supremum over finite $J$ proves ${\sum }_{s\in A_f\cap (0,t]}|\Delta f(s)|<\infty$.