View source for Jordan decomposition (of a function)

{{MSC|26A45}}

[[Category:Analysis]]

{{TEX|done}}

A canonical decomposition theorem (due to Jordan) for  [[Function of bounded variation#Jordan decomposition|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 [[Variation of a function|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 {{Cite|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 derivative|generalized derivatives]] of, respectively, $f$, $f^+$ and $f^-$ we then have that
* $\mu$ is a [[Signed measure|signed measure]]
* $\mu^+$ and $\mu^-$ are (nonnegative) measures
on the [[Borel set|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====
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|valign="top"|{{Ref|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.      {{MR|1857292}}{{ZBL|0957.49001}} 
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|valign="top"|{{Ref|Co}}|| D. L. Cohn, "Measure theory". Birkhäuser, Boston 1993.
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|valign="top"|{{Ref|Jo}}|| C. Jordan,   "Sur la série de Fourier"  ''C.R. Acad. Sci. Paris'' , '''92'''  (1881)  pp. 228–230
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|valign="top"|{{Ref|Ro}}|| H.L. Royden,   "Real analysis" , Macmillan  (1969) {{MR|0151555}} {{ZBL|0197.03501}}
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