Limit to Negative Infinity of Distribution Function of Finite Signed Borel Measure

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

Let $F_\mu$ be the distribution function of $\mu$.

Then:


 * $\ds \lim_{x \mathop \to -\infty} \map {F_\mu} x = 0$

where $\ds \lim_{x \mathop \to -\infty}$ denotes the limit at $-\infty$.

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

From Decomposition of Distribution Function of Finite Signed Borel Measure, we have:


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

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

From Limit of Distribution Function of Finite Borel Measure at Negative Infinity, we have:


 * $\ds \lim_{x \mathop \to -\infty} \map {F_{\mu^+} } x = 0$

and:


 * $\ds \lim_{x \mathop \to -\infty} \map {F_{\mu^-} } x = 0$

From Properties of Limit at Minus Infinity of Real Function: Difference Rule, we have that:


 * $\ds \lim_{x \mathop \to -\infty} \paren {\map {F_{\mu^+} } x - \map {F_{\mu^-} } x} = 0 - 0 = 0$

So:


 * $\ds \lim_{x \mathop \to -\infty} \map {F_\mu} x = 0$