Limit of Distribution Function of Finite Borel Measure at Negative Infinity
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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.
- $\ds \bigcap_{n \mathop = 1}^\infty \hointl {-\infty} {x_n} = \O$
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