Riesz-Markov-Kakutani Representation Theorem/Lemma 4

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Lemma for Riesz-Markov-Kakutani Representation Theorem

Let $\struct {X, \tau}$ be a locally compact Hausdorff space.

Let $\map {C_c} X$ be the space of continuous complex functions with compact support on $X$.

Let $\Lambda$ be a positive linear functional on $\map {C_c} X$.

There exists a $\sigma$-algebra $\MM$ over $X$ which contains the Borel $\sigma$-algebra of $\struct {X, \tau}$.

There exists a unique complete Radon measure $\mu$ on $\MM$ such that:

$\ds \forall f \in \map {C_c} X: \Lambda f = \int_X f \rd \mu$


Notation

For an open set $V \in \tau$ and a mapping $f \in \map {C_c} X$:

$f \prec V \iff \supp f \subset V$

where $\supp f$ denotes the support of $f$.


For a compact set $K \subset X$ and a mapping $f \in \map {C_c} X$:

$K \prec f \iff \forall x \in K: \map f x = 1$


Construction of $\mu$ and $\MM$

For every $V \in \tau$, define:

$\map {\mu_1} v = \sup \set {\Lambda f: f \prec V}$

For every other subset $E \subset X$, define:

$\map \mu E = \inf \set {\map \mu V: V \supset E \land V \in \tau}$

By monotonicity of $\mu_1$:

$\mu_1 = \mu {\restriction_\tau}$



Define:

$\MM_F = \set {E \subset X : \map \mu E < \infty \land \map \mu E = \sup \set {\map \mu K: K \subset E \land K \text { compact} } }$

Define:

$\MM = \set {E \subset X : \forall K \subset X \text { compact}: E \cap K \in \MM_F}$


Lemma

$\mu$ is countably additive over pairwise disjoint collections of subsets of $\MM_F$.


Proof

Let $\sequence {E_i} \in \paren {\MM_F}^\N$ be pairwise disjoint with union $E$.

Let $\map \mu E = \infty$.

Then, by countable subadditivity:

$\ds \infty = \map \mu E \le \sum_{i \mathop = 1}^\infty \map \mu {E_i}$



So:

$\ds \map \mu E = \sum_{i \mathop = 1}^\infty \map \mu E$


Suppose $\map \mu E < \infty$.

By definition of $\MM_F$, for all $\epsilon \in \R_{>0}$, for each $i$, there exists a compact $H_i \subset E_i$ such that:

$\map \mu {H_i} > \map \mu {E_i} - 2^{-i} \epsilon$

So:

\(\ds \map \mu E\) \(\ge\) \(\ds \map \mu {\bigcup_{i \mathop = 1}^\infty H_i}\)
\(\ds \) \(=\) \(\ds \sum_{i \mathop = 1}^\infty \map \mu {H_i}\) Lemma 3
\(\ds \) \(>\) \(\ds -\epsilon + \sum_{i \mathop = 1}^\infty \map \mu {E_i}\)

This holds for all $n \in \N$.

So $\mu$ is countably superadditive over pairwise disjoint collections of subsets of $\MM_F$.

Therefore, by Lemma 1, $\mu$ is countably additive over pairwise disjoint collections of subsets of $\MM_F$.

$\blacksquare$