Riesz-Markov-Kakutani Representation Theorem/Lemma 9
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 $f$.
The validity of the material on this page is questionable. In particular: The proof does not work with this definition. Something should be forgotten. Maybe, $f \prec V \iff 0 \le f \le {\mathbf 1}_V$? You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by resolving the issues. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Questionable}} from the code.If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page. |
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:
- $\map {\mu_1} V = \sup \set {\Lambda f: f \prec V}$
The validity of the material on this page is questionable. In particular: The definition of $\mu_1$ seems wrong, as $\map {\mu_1} V \in \set {0, +\infty}$ for all $V$. Indeed, for each $c>0$, $f \prec V \iff c f \prec V $. This means $\map {\mu_1} V = c \map {\mu_1} V$ for all $c > 0$. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by resolving the issues. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Questionable}} from the code.If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page. |
Note that $\mu_1$ is monotonically increasing.
That is, for all $V, W \in \tau$ such that $V \subset W$, we have:
\(\ds \map {\mu_1} V\) | \(=\) | \(\ds \sup \set {\Lambda f: \supp f \subset V}\) | ||||||||||||
\(\ds \) | \(\le\) | \(\ds \sup \set {\Lambda f: \supp f \subset W}\) | \(\ds = \map {\mu_1} W\) |
$\Box$
For every other subset $E \subset X$, define:
- $\map \mu E = \inf \set {\map {\mu_1} V: V \supset E \land V \in \tau}$
Since $\mu_1$ is monotonically increasing:
- $\mu_1 = \mu {\restriction_\tau}$
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
$\MM_F = \set {E \in \MM: \map \mu E < \infty}$
Proof
- $E \in \MM_F \implies \paren {\forall K \text{ compact}: E \cap K \in \MM_F \implies E \in \MM}$
That is:
- $\MM_F \subset \MM$
Conversely, suppose that:
- $E \in \MM$
- $\map \mu E < \infty$
- $\epsilon \in \R{>0}$
By definition of $\mu$, there exists an open $V \supset E$ such that:
- $\map \mu V < \infty$
By Lemma 5 and Lemma 6, there exists a compact $K \subset V$ such that:
- $\map \mu {V \setminus K} < \epsilon$
By definition of $\MM$:
- $E \cap K \in \MM_F$.
So by definition of $\MM_F$, there exists a compact $H \subset E \cap K$ such that:
- $\map \mu {E \cap K} < \map \mu H + \epsilon$
Since $E \subset \paren {E \cap K} \cup \paren {V \setminus K}$, by Lemma 1:
- $\map \mu E \le \map \mu {E \cap K} + \map \mu {V \setminus K} + 2 \epsilon$
Thus:
- $E \in \MM_F$.
Therefore:
- $\MM_F = \set {E \in \MM: \map \mu E < \infty}$
$\blacksquare$