Distribution Function of Finite Borel Measure is Right-Continuous

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Theorem

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

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


Then $F_\mu$ is right-continuous.


Proof

From Sequential Right-Continuity is Equivalent to Right-Continuity in the Reals: Corollary, it suffices to show that:

for all monotone sequences $\sequence {x_n}_{n \mathop \in \N}$, with $x_n > x$ for each $n$, that converge to $x$ we have:
$\ds \map {F_\mu} x = \lim_{n \mathop \to \infty} \map {F_\mu} {x_n}$

Now let $\sequence {x_n}_{n \mathop \in \N}$ be a monotone sequence with $x_n > x$ for each $n \in \N$ and $x_n \to x$.

Then $\sequence {\hointl {-\infty} {x_n} }_{n \mathop \in \N}$ is a decreasing sequence of sets.

Then from Limit of Decreasing Sequence of Unbounded Below Closed Intervals, we have:

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

Since $\mu$ is finite, we have:

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

so Measure of Limit of Decreasing Sequence of Measurable Sets is admissable and gives:

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

That is:

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

so:

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

Hence:

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

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

$F_\mu$ is right-continuous.

from Sequential Right-Continuity is Equivalent to Right-Continuity in the Reals: Corollary.

$\blacksquare$


Sources

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