Integral Representation of Linear Functionals on Function Spaces: Reisz-Markov Theorem
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Keywords:
Boral measure, $\mu$-measurable, linear functional, $\sigma$-algebraAbstract
Let $\mu$ be a regular Borel measure on X, where X is locally compact Hausdrof space. Let $\phi$ be defined on $\mathcal{L}(X)$ such that $\phi(f)=\int f d\mu$, where $f\in\mathcal{L}(X)$. $\phi(\alpha f+\beta g)=\int(\alpha f+\beta g)d\mu= \alpha\int f d\mu +\beta\int g d\mu=\alpha\phi(f)+ \beta\phi(g)$ then $\phi$ is a positive linear functional on $\mathcal{L}(X)$. Thus every regular Borel measure defines a positive linear functional on $\mathcal{L}(X)$. Where $\mathcal{L}(X)$ is the $\alpha$-algebra of $\mu$-measurable functions on X. Here we wish to discuss the converse of this, that for every Positive linear functional $\phi$ on $\mathcal{L}(X)$ there exist a unique regular Borel measure on X such that $\phi(f)=\int f d\mu$, $\forall f\in \mathcal{L}(X)$. The result is known as Reisz Markov Theorem.
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