Riesz-Markov-Kakutani Representation Theorem/Notation
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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 used
The purpose of this page is to document the notation used for the proof of the Riesz-Markov-Kakutani Representation Theorem.
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 $f \in \map {C_c} X$:
- $K \prec f \iff \forall x \in K: \map f x = 1$