# 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 "Borel structures for function spaces." 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 "On choosing a function at random." 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).

• I suspected I would get a great answer from you, thank you! On a related note, if we considered the set $A_2^{\Delta{A_1}}$, could I potentially use Tychonoff to view this as a compact set perhaps? – Walrasian Auctioneer Apr 1 at 20:15
• Yes, but there are some problems with this approach. First, measurable functions do not form a closed subset. So limits will generally not be measurable. Second, in general topological spaces, compactness need not imply sequential compactness and here it does not. Maybe I can say something more positive if you give some detail on what you are trying to achieve. Games like that have certainly been studied. – Michael Greinecker Apr 1 at 20:21
• I'm interested in looking at limits of finite approximations of extensive form games, and their resulting strategies as potential equilibria of the limit game, ala Harris, Reny, Robson etc. It seems straightforward that strategies of the first stage players do not run into issues, but convergence of continuation strategies seem to be ill-defined as you mentioned. – Walrasian Auctioneer Apr 1 at 20:28
• I see. Myerson and Reny tried something similar for their recent paper on an extensive-form refinement, but they had to dispose of limiting strategies completely; only the final distributions over outcomes was defined. – Michael Greinecker Apr 1 at 20:33
• My question was motivated from that very paper! I better stop comments before the mods get angry for being too chatty, but thanks for the answer. – Walrasian Auctioneer Apr 1 at 20:37