Notation question for probability measures on product spaces

  • #1
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Let [itex](A,\mathcal A), (B,\mathcal B)[/itex] be measurable spaces. Let [itex]p[/itex] be a probability measure on [itex](A,\mathcal A)[/itex], and let [itex]q:A\to\mathcal P(B,\mathcal B)[/itex] be a measurable function which takes each [itex]a\in A[/itex] to some probability measure [itex]q(\cdot|a)[/itex] on [itex](B,\mathcal B).[/itex] Then there is a unique probability measure [itex]\mu[/itex] on [itex](A\times B, \mathcal A\otimes\mathcal B)[/itex] which has [tex]\mu(\hat A\times \hat B) = \int_{\hat A} q(\hat B|\cdot)\text{ d}p[/tex] for every [itex]\hat A\in\mathcal A, \hat B\in\mathcal B.[/itex]

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

Answers and Replies

  • #2
18,482
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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?
 
  • #3
269
24
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.
 

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