View source for Signed measure

{{MSC|28A33}}

[[Category:Classical measure theory]]

{{TEX|done}}

''generalized measure'', ''real valued measure''

$\newcommand{\abs}[1]{\left|#1\right|}$
An real-valued $\sigma$-additive function defined on a certain $\sigma$-algebra $\mathcal{B}$ of subsets of
a set $X$. More generally one can consider vector-valued measures, i.e. $\sigma$-additive functions $\mu$ on $\mathcal{B}$
taking values on a Banach space $B$ (see [[Vector measure]]). The total variation measure of $\mu$ is defined on $B\in\mathcal{B}$ as:
\[
\abs{\mu}(B) :=\sup\left\{ \sum \abs{\mu(B_i)}_B: \text{$\{B_i\}\subset\mathcal{B}$ is a countable partition of $B$}\right\}
\]
where $\abs{\cdot}_B$ denotes the norm of $B$.
In the real-valued case the above definition simplifies as
\[
\abs{\mu}(B) = \sup_{A\in \mathcal{B}, A\subset B} \left(\abs{\mu (A)} + \abs{\mu (X\setminus B)}\right).
\]
$\abs{\mu}$ is a measure and $\mu$ is said to have finite total variation if $\abs{\mu} (X) <\infty$. 

If $V$ is finite-dimensional the [[Radon-Nikodym theorem]] implies the existence
of a measurable $f\in L^1 (\abs{\mu}, V)$ such that
\[
\mu (B) = \int_B f d\abs{\mu}\qquad \mbox{for all $B\in\mathcal{B}$.}
\]
In the case of real-valued measures this implies that each such $\mu$ can be written as the difference
of two nonnegative measures $\mu^+$ and $\mu^-$ which are mutually singular (i.e. such that there are
sets $B^+, B^-\in\mathcal{B}$ with $\mu^+ (X\setminus B^+)= \mu^- (X\setminus B^-)
=\mu^+ (B^-)=\mu^- (B^+)=0$). This last statement is sometimes referred to as Hahn decomposition theorem.

By the [[Riesz representation theorem]] the space of signed measures with finite total 
variation on the Borel $\sigma$-algebra of a locally compact
Hausdorff space is the dual of the space of continuous functions (cp. also with [[Convergence of measures]]).

====References====
{|
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|valign="top"|{{Ref|AmFuPa}}||  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}} 
|-
|valign="top"|{{Ref|Bo}}||  N. Bourbaki, "Elements of mathematics. Integration" , Addison-Wesley  (1975) pp. Chapt.6;7;8 (Translated from French) {{MR|0583191}}  {{ZBL|1116.28002}} {{ZBL|1106.46005}} {{ZBL|1106.46006}}  {{ZBL|1182.28002}} {{ZBL|1182.28001}} {{ZBL|1095.28002}}  {{ZBL|1095.28001}} {{ZBL|0156.06001}}
|-
|valign="top"|{{Ref|DS}}||  N. Dunford, J.T. Schwartz, "Linear operators. General theory" , '''1'''  , Interscience (1958) {{MR|0117523}}
|-
|valign="top"|{{Ref|Bi}}||  P. Billingsley, "Convergence of probability measures" , Wiley (1968)  {{MR|0233396}} {{ZBL|0172.21201}}
|-
|valign="top"|{{Ref|Ma}}||  P. Mattila, "Geometry of sets and measures in euclidean spaces.  Cambridge Studies in Advanced Mathematics, 44. Cambridge University  Press, Cambridge,  1995. {{MR|1333890}} {{ZBL|0911.28005}}
|-
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