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?
Best Answer
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.