Jordan decomposition (of a function)

2020 Mathematics Subject Classification: Primary: 26A45 [MSN][ZBL]

A canonical decomposition theorem (due to Jordan) for functions of bounded variation of one real variable.

Theorem If $f:[a,b] \to\mathbb R$ is a function of bounded variation then there is a pair of nondecreasing functions $f^+$ and $f^-$ such that $f= f^+- f^-$ and $TV (f) = f^+ (b)-f^+ (a) + f^- (b)- f^- (a)$ (where $TV (f)$ denotes the total variation of $f$ (see also Function of bounded variation). The pair is unique up to addition of a constant, i.e. if $g^+$ and $g^-$ is a second pair with the same property, then $g^+-f^+=g^--f^-\equiv {\rm const}$.

For a proof see Section 5.2 of [Ro].

The decomposition is related to the Jordan decomposition (of a signed measure). More precisely, if we denote by $\mu$, $\mu^+$ and $\mu^-$ the generalized derivatives of, respectively, $f$, $f^+$ and $f^-$ we then have that

• $\mu$ is a signed measure
• $\mu^+$ and $\mu^-$ are (nonnegative) measures

on the Borel sets of $\mathbb R$ and $\mu = \mu^+-\mu^-$ is the Jordan decomposition of $\mu$. For more details we refer to Function of bounded variation.

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