Naming and notation for joint measures with marginal and conditional densities

[economicsnerd]

Let $(A,\mathcal A), (B,\mathcal B)$ be measurable spaces. Let $p$ be a probability measure on $(A,\mathcal A)$, and let $q:A\to\mathcal P(B,\mathcal B)$ be a measurable function which takes each $a\in A$ to some probability measure $q(\cdot|a)$ on $(B,\mathcal B).$ Then there is a unique probability measure $\mu$ on $(A\times B, \mathcal A\otimes\mathcal B)$ which has $$\mu(\hat A\times \hat B) = \int_{\hat A} q(\hat B|\cdot)\text{ d}p$$ for every $\hat A\in\mathcal A, \hat B\in\mathcal B.$

The question: Is there a typical thing to call $\mu$? Does it have a name, in terms of $p$ and $q$? How about notation? $pq$? $p\otimes q$ (which would be misleading)? $q\circ p$? $q^p$? I looked around and couldn't find anything consistent.

 

[Greg Bernhardt]

I'm sorry you are not generating any responses at the moment. Is there any additional information you can share with us? Any new findings?

 

[economicsnerd]

No such luck. In the thing I'm writing, I just named it $\mu_{p,q}$ and fully defined it, since I couldn't find a standard name for it. I figured I'll fix it later if I stumble on a good name elsewhere.