Topology on the space of measurable functions The context is as follows:
Suppose we have a 2 period sequential game, with player $i$ in stage $i$, with action set $A_i$.
Give $A_i$ all the nice properties, as compact, separable metric spaces (I'd be happy to even consider $A_i \subset \mathbb{R})$.
A strategy for player 1 is a map $\sigma \in \Delta(A_1)$, and for P2 a measurable function $\eta:A_1 \rightarrow \Delta(A_2)$.
Its clear that when I endow $\Delta(A_1)$ with the weak topology, I can talk about a sequence of strategies having a convergent subsequence.
Is there a topology I can put on $\Sigma=\{\eta:A_1 \rightarrow \Delta(A_2) : \eta \text{ measurable}\}$ that allows me to extract convergent subsequences?
Not really. There are many compact metrizable topologies you can put on this space, but none that relate meaningfully to the structure of the problem.
Let's look first at the case $A_1=[0,1]$ and $A_2=\{0,1\}$. Consider the elevation function $e:A_1\times\Sigma\to\Delta(A_2)=[0,1]$ given by $e(a,\eta)=\eta(a)$. If you want the ultimate action choice of player $2$ to be continuous in player $1$'s action and the strategy choice of player $2$, you need $e$ to be continuous. In particular, $e$ should be measurable when you endow $\Sigma$ with the Borel $\sigma$-algebra. That never works:
It follows from the main result (Theorem D) in Robert Aumann's 1961 paper &quot;Borel structures for function spaces.&quot; that there is no Borel $\sigma$-algebra on $\Sigma$ that makes $e$ jointly measurable. For consumers,  I would recommend to read Aumann's 1963 article &quot;On choosing a function at random.&quot; instead.
To make this a bit more digestible, let $F$ be the set of measurable functions from the unit interval to itself. Let $e:[0,1]\times F\to [0,1]$ be given by $e(x,f)=f(x)$. If we endow $[0,1]$ with the Borel $\sigma$-algebra and $F$ with any (!) $\sigma$-algebra, $e$ will necessarily be nonmeasurble. Now, by Kuratowski's isomorphism theorem, all uncountable separable and complete metric spaces are isomorphic as measurable spaces. This will include $A_1$ (if uncountable) and $\Delta(A_2)$ (if $A_2$ has at least $2$ elements).