Jordan decomposition lemma (bounded variation)

Let $f:[a,b]\to\mathbb{R}$ be a function of bounded variation . For $x\in[a,b]$, write $V_a^x(f)=\sup\left\{\sum_{j=1}^n |f(t_j)-f(t_{j-1})|:\ a=t_0<t_1<\cdots<t_n=x\right\}$ for the total variation of $f$ on $[a,x]$.

Jordan decomposition lemma: If $f$ has bounded variation on $[a,b]$, then there exist increasing functions $g,h:[a,b]\to\mathbb{R}$ such that $f=g-h.$ One explicit choice is $h(x)=V_a^x(f),\qquad g(x)=f(x)+V_a^x(f).$ Then $h$ is increasing, and $g$ is increasing as well.

This lemma is fundamental for the Riemann–Stieltjes integral : bounded-variation integrators behave like differences of increasing (hence “measure-like”) functions.

Proof sketch: The map $x\mapsto V_a^x(f)$ is increasing by definition. For $a\le x<y\le b$, the variation estimate implies $V_a^y(f)-V_a^x(f)\ge |f(y)-f(x)|\ge -(f(y)-f(x)),$ so $g(y)-g(x)=(f(y)-f(x))+\bigl(V_a^y(f)-V_a^x(f)\bigr)\ge 0,$ hence $g$ is increasing. Finally, $f=g-h$ holds by construction.