"Nice" sigma-algebra on set of measurable functions In topology, given topological spaces $X$ and $Y$, the compact-open topology is considered, under the relatively mild requirement that $X$ be locally compact Hausdorff, to be the most "natural" topology on the set $\mathcal{C}(X,Y)$ of continuous functions $X\to Y$. (I'm not going to even attempt to define "natural" here---take it to mean whatever seems most appropriate).
Is there something analogous, in the sense of being somehow "natural", for the set of measurable functions between two measure spaces?
 A: There is the Effros Borel structure.  But that is a $\sigma$-algebra for the collection $F(S)$ of closed sets in a Polish space $S$.  But it is again a standard Borel structure, so maybe it meets your criterion of "nice"...  
Google "effros borel structure" for more information.
A: There is an impossibility theorem: If you let $\mathcal{L}$ be the the space of Borel-measurable functions $f:[0,1]\to[0,1]$, and $e:\mathcal{L}\times [0,1]\to[0,1]$ the evaluation given by $e(f,x)\mapsto f(x)$, then there is no $\sigma$-algebra on $\mathcal{L}$ such that the evaluation is jointly measurable. The result is a consequence of the rather complicated classification result in  R. Aumann, Borel Structures for Function Spaces, Illinois Journal of Mathematics 5 (1961), pp. 614-630. Easier proofs of the main results can be found in the paper "Borel Structures for Function Spaces" (yes, same title) by B.V. Rao, Colloquium Mathematicum, 1971.
A $\sigma$-algebra on measurable functions I have actually seen used is the following: If $(S,\mathcal{S})$ and $(T,\mathcal{T})$ are measurable spaces, we endow the family of measurable functions between them with the $\sigma$-algebra generated by sets of the form $\{f:f(s)\in B\}$ with $s\in S$ and $B\in\mathcal{T}$. The author used this $\sigma$-algebra to show that to each Markov kernel from $S$ to $T$, there corresponds a certain probability measure on this $\sigma$-algebra. The paper is H. v. Weizsäcker Zur Gleichwertigkeit zweier Arten der Randomisierung, Manuscripta Mathematica 11 (1974).
