Let $\text{Meas}$ be the category of measurable spaces with measurable functions as morphisms. Does $\text{Meas}$ have [exponential objects](https://en.wikipedia.org/wiki/Exponential_object)?



Exponential objects in the category of measurable spaces
As mentioned in the comments, Meas of course has *some* exponential objects $B^A$, but not for all $A$ and $B$, i.e., it is not cartesian closed.
This fact is discussed as Proposition 6 of [A Convenient Category for Higher-Order Probability Theory](http://arxiv.org/pdf/1701.02547.pdf) by Heunen, Kammar, Staton, and Yang, citing an old result of Aumann:


* R. J. Aumann, "Borel structures for function spaces," *Illinois Journal of Mathematics*, vol. 5, pp. 614–630, 1961. [project euclid](https://projecteuclid.org/euclid.ijm/1255631584)


