\$ 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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## 1 prize 1

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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

TeunTeun
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