= Cantor-space representation of positive functionals
{c}
{title2=$\phi(f)=\int_\Omega f\,dP$}
Extend the <cylinder premeasure from a positive functional> to a <Borel probability measure> and use <cylinder-function density in Cantor space>. The integral and functional agree on cylinder <simple functions>, and both are continuous in the <supremum norm>, so they agree on all of $C(\Omega)$. Uniqueness on the generating cylinder algebra gives uniqueness of the Borel <measure>.
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