When is a space of measures a measurable space? Let $X$ denote a measurable space, that is, a set equipped with a $\sigma$-algebra $\Sigma(X)$. Let $M(X)$ denote the space of real-valued measures over $X$. This is a vector space over the real numbers, since the sum of two measures is again a measure, as is a scalar multiple of a measure. I would like to know the most general setting for which $M(X)$ is a measurable vector space.
Does $M(X)$ admit a canonical choice of $\sigma$-algebra, turning it into a measurable space?
If the answer is "no", then what about the setting where $X$ is a localizable measurable space?
If the answer is again "no", then what is the most general setting so that $M(X)$ is admits a canonical measurable structure?
Most generally, what is the largest subcategory $\mathcal C$ of $\mathbf{Meas}$ so that $M : \mathcal C \to \mathcal C$ is an endofunctor?
 A: Let $(X,\Sigma)$ be the measurable space.  I think the sigma-algebra on $\mathcal M$ that you want is this.  The least sigma-algebra so that for all $A \in \Sigma$, the map $\mu \mapsto \mu(A)$ is measurable.
A: First try it with $X$ a point. Then the space of measures is $\mathbb R^+$.
If you want the measure to be bounded, you give up at this point. I think the right thing to do here is to look at the space of probability measures. Then on a finite set of points you get a simplex, which has a canonical measure. However, once you set $X = \mathbb N$ (with the discrete $\sigma$-algebra, which is also the unique $\sigma$-\algebra that separates the points), it is clear that there cannot be a natural probability measure on the space of probability measures, because it would have to be invariant under the permutation action, so the expected size of each point must be equal, which is clearly impossible.
So you have to give up boundedness -some sets of measures will have infinite measure. For a single point, Lebesgue measure is the obvious choice. For a finite set of points, Lebesgue measure still makes sense. For an infinite set of points, I have no idea what to do - I'm not even sure which sets of measures should have infinite measure.
