Jordan decomposition lemma

Jordan decomposition lemma: Let $a<b$ and let $g:[a,b]\to\mathbb{R}$ be a bounded variation function . Then there exist increasing functions $g_1,g_2:[a,b]\to\mathbb{R}$ such that

Moreover, one can choose $g_1,g_2$ so that the total variation satisfies

This decomposition reduces many questions about bounded variation to the monotone case, and it is frequently used in the theory of the Riemann–Stieltjes integral .