User:Caliburn/s/mt/Measure of Singleton under Finite Borel Measure in terms of Distribution Function

Theorem

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

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

Let $b \in \R$.

Then:

$\map \mu {\set b} = \map {F_\mu} b - \map {F_\mu} {b^-}$

where:

$\ds \map {F_\mu} {b^-} = \lim_{x \mathop \to b^-} \map {F_\mu} x$, where $\ds \lim_{x \mathop \to b^-}$ denotes the left limit at $b$.

Proof

From Distribution Function of Finite Borel Measure is Increasing, we have:

$F_\mu$ is increasing.

From Distribution Function of Finite Borel Measure is Bounded, we have:

$F_\mu$ is bounded.

So, from Limit of Increasing Function, we have:

$\ds \lim_{x \mathop \to b^-} \map {F_\mu} x$ exists.

Write:

$\ds \map {F_\mu} {b^-} = \lim_{x \mathop \to b^-} \map {F_\mu} x$

From Sequential Characterization of Left-Limit of Real Function: Corollary, we then have:

for all monotone sequences $\sequence {x_n}_{n \mathop \in \N}$ with $x_n < b$ for each $n \in \N$ and $x_n \to b$, we have:

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

Sources

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