# Notation question for probability measures on product spaces

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.

## Answers and Replies

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?

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.