Suppose we have a measure space $(X,a)$, a measurable space $Y$ and for every $x\in X$ we have a measure $b_x$ on $Y$. Suppose that $(Z,c)$ is a measure space such that as a measurable space $Z=X\times Y$ and the measure $c$ is the integral of measures $b_x$ with respect to the measure $a$.

I have a vague intuition that $(Z,c)$ is a coproduct of the family $(Y,b_x)$ along $(X, a)$

Question: Is there a category of measure spaces and a categorical construction in it which captures this intuition?

One could ask a similar question about a category of Hilbert spaces and integrals of Hilbert spaces over measure spaces, and preferably the "categorical construction" in question should also answer this problem.

The categorical construction in question should preferably be similar to coproduct, and it would be very nice if it actually was a coproduct in some category.

On the other hand, maybe my vague intuition is wrong, and comments on that would also be appreciated.