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:

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

$\blacksquare$

Sources

• 2013: Donald L. Cohn: Measure Theory (2nd ed.) ... (previous) ... (next): $4.4$: Functions of Finite Variation