Limit of Distribution Function of Finite Borel Measure at Negative Infinity

Theorem

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

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

Then:

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

Proof

From Sequential Characterisation of Limit at Minus Infinity of Real Function: Corollary, we aim to show that:

for all decreasing real sequences $\sequence {x_n}_{n \mathop \in \N}$ with $x_n \to -\infty$ we have $\ds \lim_{n \mathop \to \infty} \map {F_\mu} {x_n} = 0$.

Since $\sequence {x_n}_{n \mathop \in \N}$ is decreasing, we have:

the sequence $\sequence {\hointl {-\infty} {x_n} }_{n \mathop \in \N}$ is decreasing.

From Limit of Decreasing Sequence of Unbounded Below Closed Intervals with Endpoint Tending to Negative Infinity:

$\ds \bigcap_{n \mathop = 1}^\infty \hointl {-\infty} {x_n} = \O$

Further, since $\mu$ is a finite measure, we have:

$\ds \map \mu {\hointl {-\infty} {x_1} } < \infty$

So Measure of Limit of Decreasing Sequence of Measurable Sets is admissible, and gives:

$\ds \map \mu {\bigcap_{n \mathop = 1}^\infty \hointl {-\infty} {x_n} } = \lim_{n \mathop \to \infty} \map {F_\mu} {x_n}$

Since:

$\ds \bigcap_{n \mathop = 1}^\infty \hointl {-\infty} {x_n} = \O$

we have:

$\ds \map \mu {\bigcap_{n \mathop = 1}^\infty \hointl {-\infty} {x_n} } = 0$

from Empty Set is Null Set.

So:

$\ds \lim_{n \mathop \to \infty} \map {F_\mu} {x_n} = 0$

Since the sequence $\sequence {x_n}_{n \mathop \in \N}$ was arbitrary, we have:

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

by Sequential Characterisation of Limit at Minus Infinity of Real Function: Corollary.

$\blacksquare$

Sources

• 2013: Donald L. Cohn: Measure Theory (2nd ed.) ... (previous) ... (next): $1.3$: Outer Measures