Decomposition of Distribution Function of Finite Signed Borel Measure

Theorem

Let $\mu$ be a finite signed Borel measure on $\R$.

Let $\tuple {\mu^+, \mu^-}$ be the Jordan decomposition of $\mu$.

Then:

$F_\mu = F_{\mu^+} - F_{\mu^-}$

where:

$F_\mu$ is the distribution function of $\mu$

$F_{\mu^+}$ and $F_{\mu^-}$ are the distribution functions of $\mu^+$ and $\mu^-$ respectively.

Proof

For each $x \in \R$ we have:

\(\ds \map {F_\mu} x\)

\(=\)

\(\ds \map \mu {\hointl {-\infty} x}\)

Definition of Distribution Function of Finite Signed Borel Measure

\(\ds \)

\(=\)

\(\ds \map {\mu^+} {\hointl {-\infty} x} - \map {\mu^-} {\hointl {-\infty} x}\)

Definition of Jordan Decomposition

\(\ds \)

\(=\)

\(\ds \map {F_{\mu^+} } x - \map {F_{\mu^-} } x\)

Definition of Distribution Function of Finite Borel Measure

$\blacksquare$