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.

References

[AFP]	L. Ambrosio, N. Fusco, D. Pallara, "Functions of bounded variations and free discontinuity problems". Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York, 2000. MR1857292Zbl 0957.49001
[Co]	D. L. Cohn, "Measure theory". Birkhäuser, Boston 1993.
[Jo]	C. Jordan, "Sur la série de Fourier" C.R. Acad. Sci. Paris , 92 (1881) pp. 228–230
[Ro]	H.L. Royden, "Real analysis" , Macmillan (1969) MR0151555 Zbl 0197.03501