Notation question for probability measures on product spaces

[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.