$ ildePhi(mu, u) dot= int_A imes BPhi(a, b) ubraintv-jp.comrmdmu otimes ubraintv-jp.comrmd u$

In a context where:

$A$ and $B$ are compact metric spaces$mu$ and also $ u$ are probability distribution over $A$ and also $B$, resp.$Phi$ is a constant function $A imes B ightarrow ubraintv-jp.combbR$$ ildePhi(mu, u)$ is stated to it is in the*expected value*the $Phi$

I recognize that you need to incorporate over $A imes B$ to obtain this intended value, and also to take $mu$ and also $
u$ distributions into account while doing this. But..

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How to be I claimed to recognize the $otimes$ symbol here? What is this operation? how does $ubraintv-jp.comrmdmu$ relates come $a$ and $ubraintv-jp.comrmd u$ relates to $b$ within this integrand?

(To obtain the full context, I"ve uncovered this in this pretty neat notes introducing differential game theory (equation 2.8 web page 13).)

integration probability-distributions notation

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asked Feb 15 "17 in ~ 16:12

iago-litoiago-lito

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I assume $mu$ and $ u$ are distributions with $dmu$ and also $d u$ the corresponding measures. Climate $dmuotimes d u$ denotes the product measure. Distribution $mu$ have the right to be viewed as generalised functions, where $mu(x)$ doesn"t need to be defined. However, i think in your situation $mu$ and also $ u$ are just functions, for this reason you might write $mu(x)$ and also $ u(y)$. In this situation $dmu(x)=mu(x)dx$, and the product measure becomes simply$$(dmuotimes d u)(a,b)=mu(a) u(b)dadb,.$$Now, to it is in (overly) precise, the author actually intended to write$$ ildePhi:=int_A imes BPhi(a,b)(dmuotimes d u)(a,b),,$$but he assumed that it to be clear that $a$ and also $b$ are integrated over.So once $mu$ and $ u$ space functions, friend have$$ ildePhi=int_A imes BPhi(a,b)mu(a) u(b)dadb,.$$

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answered Jul 2 "19 at 10:10

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