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: