I am curious about how we can get the existence of product measures in the infinite dimensional space by using the Riesz representation theorem.
I am studying the Kolmogorov extension theorem by the book of Tao, An introduction to measure theory. At the page 241, he says, "This result can also be obtained from the Riesz representation theorem, which is covered in 1.10 of An epsilon of room, Vol. I.", where 'This result' refers to the existence of product measures in the infinite dimensional space. I searched the book he mentioned and there was only the result of Riesz thm, not the proof I want.
Here is the statement of the existence of product measures
Let A be an arbitrary set. For each $\alpha\in A$, let $(X_\alpha, \mathcal{B}_\alpha, \mu_\alpha)$ be a probability space in which $X_\alpha$ is a locally comapct, $\sigma$-comapct metric space, with $\mathcal{B}_\alpha$ being its Borel $\sigma$-algebra (i.e. the $\sigma$-algebra generated by the open sets). Then there exists a unique probability measure $\mu_\alpha$=$\Pi_{\alpha\in A}\mu_\alpha$ on $(X_A, \mathcal{B}_A)$:=$(\Pi_{\alpha\in A}X_\alpha, \Pi_{\alpha\in A}\mathcal{B}_\alpha)$ with the property that
$\mu_A(\Pi_{\alpha\in A}E_\alpha)=\Pi_{\alpha\in A}\mu_\alpha(E_\alpha)\\ \\$
whenever $E_\alpha\in \mathcal{B}_\alpha$ for each $\alpha\in A$, and one has $E_\alpha=X_\alpha$ for all but finitely many of the $\alpha$
I considered at first that some construction of directed system of measure spaces would yield a desired linear functional as the inverse limit, but I noticed that even if I can get the linear functional, the Riesz theorem is not applicable because the space $X_A$ need not be locally compact when A is infinite.
More precisely, I considered the canonical projection map $\pi^B_C:X_B\rightarrow X_C$ whenever $C\subset B$ and B, C finite index sets, and we may get the pull back map on $C_c(X_C)$ to $C_c(X_B)$ by taking a composition inside, and again we may take the push forward map from $C_c(X_B)^*$ to $C_c(X_C)^*$, and we have the directed system of dual spaces. I'm not sure whether my reasoning is correct.
But anyway I'm stuck with the problem that X_A need not be locally coampact when A is infinite, since infinite product of locally compact spaces need not be so.(Maybe it is the problem of the Topology by Munkres) How can you solve this problem?