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Also the space $\beta\mathbb{N}\setminus\mathbb{N}$ satisfies the almost P-space property. This is explicit, for instance, in Hindman and Strauss's book MR1642231, Theorem 3.36.
Well, I ask for compact. If we go for the weak topology (and I'm not even sure that in non-separable spaces that still gives a compact unit ball) then I don't see right away why any open set has unaccountably many (disjoint?) open subsets
After Bill's comment I arrived at a similar proof, but was too lazy to write it here. This answer gives good amount of details so I will go ahead and accept it (and I learned about invariance of domain!).
I didn't understand what $X$ and $x$ are. Real numbers? $\pm$ states? I also didn't understand what is $p_+$ (or $p_-$). Is it the distribution of $X$ and does $X$ depend on time $t$?
It seems to me that you can model this with (directed) graphs instead. To be precise, make the set of vertices be $W$ and you have an edge from $i$ to $j$ if $j\in f(i)$. Then you want to know the expectation of the size of "the set you can get to, starting on $V$". This seems more natural to me. Unfortunately, either way I don't have a clue, unless we know more about the distribution of $f$ (or in this formulation of the edges)
