I am trying to get the implications of the following general setting concerning compact spaces and continuous maps.
Any feedback would be greatly appreciated, because I have some difficulties in applying what I know from the books, to settings that are outside books.
Assume that $X, Y$ are compact metric spaces, and that there is a map
$$ \mu : X \to \Delta (X \times Y)$$
such that $\mu$ is continuous, where $\Delta (\Omega)$ denotes the set of probability measures over a generic $\Omega$ . Endow $X \times Y$ with the product topology, and $ \Delta (X \times Y)$ with the topology of weak convergence.
Broad Problems:
What should I think exactly when I have such a setting?
What conclusions should I infer regarding it?
Some thoughts & a more specific problem:
The continuity of $\mu$ tells us that when we have an open (resp. closed) subset $G$ of $ \Delta (X \times Y)$, we are ensured that the preimage $\mu^{-1} (G)$ is open (resp. closed). But here my intuition is close to failure and I have the following problems.
- Let $X$ be finite, hence compact. Let $G = \{ \delta_{(x,y)} \}$, where $\delta$ denotes the Dirac measure for some elements $x \in X$ and $y \in Y$. Thus $\mu^{-1} (\{ \delta_{(x,y)} \} )$ maps to some element $ x \in X$. But now, how can $\mu$ really be continuous in this case?
Do we actually base the statement on the topological notion of continuity, based in this case on the idea that we are implicitly endowing the finite $X$ with the discrete topology? (Problem 1) - Is it correct the fact that a subset of $ \Delta (X \times Y)$ is actually a number? (it should be the case, but better to be sure...) (Problem 2)
- Is it better to see at Problem 1 as I stated, but focusing on closed sets, rather than open. Indeed, we are allowed, because in the discrete topology, subsets are clopen, and a specific number attached to $\mu$ should be closed in the topology of weak convergence, e.g. $G = 1/2$.
Is this a correct line of reasoning? (Problem 3)