I couldn't find this on math.se or by searching the internet. Thanks for any help!
OK, so I have a set $\Omega$. For now, let's think of $\Omega$ as finite, consisting of $n$ elements. Now consider the set $S$ of probability measures on $\Omega$ (or equivalently just the probability simplex in $\mathbb{R}^n$). I am interested in the space of square-integrable functions from $S$ to $\mathbb{R}$. That is, I am interested in $L^2[S,\mu]$ where $\mu$ is uniform on $S$ (i.e. think of $S$ as the probability simplex in $\mathbb{R}^n$ and $\mu$ as the Lebesgue measure on it).
Goal: I would like to construct an orthonormal basis for $L^2[S,\mu]$. (For the curious: The end goal is to write a generalized Fourier series for any given function $f: S \to \mathbb{R}$.)
Question 1: What possible orthonormal bases might I choose, and is there a particularly "nice" one? I think that I could just take the standard basis for $[0,1]^n$ (see this question) and that would technically work. (Right?) But I was wondering if there is some nicer basis that is more natural in this setting --- maybe one that relates to the fact that elements of $S$ are probability measures.
Question 2: Can I extend this question meaningfully to the case where $\Omega$ is not necessarily finite, so $S$ is an arbitrary probability measure space? That is, is there some canonical $\mu$ so that $L^2[S,\mu]$ exists, and then, can we construct some explicit orthonormal basis? The $\mathbb{R}^n$ trick shouldn't work any more....
Thanks again!