It depends. What Aumann is concerned with in the paper are admissible structures, that is, he looks for a sigma-algebras on sets of measurable functions $F\subseteq Y^X$ such that th evaluation $e:Y^X\times X\to Y$ given by $e(f,x)=f(x)$ is measurable with respect to the product-sigma-algebra.
A well behaved case is when, say, $X=Y=\mathbb{R}$ X=[0,1]$, $Y=\mathbb{R}$ with the usual topology. Then $C(A,B)$ C(X,Y)$ is a complete, separable metric space when endowed with the uniform metric and the evaluation function is continuous, hence measurable when $C(A,B)$ C(X,Y)$ is endowed with the Borel sigma-algebra. The discrete sigma-algebra is even larger, so the evaluation is certainly measurable in this case.
Now let $X=Y$ be discrete topological spaces with cardinality larger than the continuum. Let $y\in Y$. We show that $e^{-1}(\{y\})$ is not in the product-sigma-algebra. Suppose it is. By the first two lemmata here, $e^{-1}(\{y\})$ has to be the union of continuum many product sets of the form $F'\times X'$, $F\subseteq Y^X$ and $X'\subseteq X$. For such a product set, we have that every $f\in F'$ is constant and equal to $y$ on $X'$. W.l.o.g. we can assume that none of the $X'$ is empty or equal to $X$. So we can construct an $X''\subseteq X$ such that $X''$ intersects every $X'$ and every $(X')^C$. Let $F''=\{f\in Y^X:f(x)=y\text{ iff }x\in X''\}$. Then $F''\times X''\subseteq e^{-1}(\{y\})$ but $F''\times X''$ is not a subset of $\bigcup F'\times X'$.
