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I need help to translate into a conditional exepectation the following problem:

We have an interval of $\mathbb{R}$ of size M (say $\mathcal{M} = [0, M]$). For each element $x \in \mathcal{M}$ I define $\theta(x)$ which is the score of each element of my interval. I want these scores to follow a given distribution with cdf $F$. For each subset $\mu$ of $\mathcal{M}$, I would like to define the average value of $\theta$ over this subset.

$$ \mathbb{E}\left[\theta(x) \left|x \in \mu \right. \right] $$

How can I calculate this? My first intuition is to do

$$ \int_{x \in \mu(x)}{\theta(x) dF(x)} $$

But it just seems like I am mixing different measures. I would really appreciate some help to formalize this and relate this to the theory of probability.

Thank you for your help.

T.

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    $\begingroup$ The average value of the function $\theta:x\mapsto\theta(x)$ over the set $\mu\subseteq\mathcal M$ of positive measure $|\mu|$ is by definition $$\frac1{|\mu|}\int_\mu\theta(x)\,dx$$ Nothing probabilistic in there. $\endgroup$ – Did Dec 7 '18 at 19:22
  • $\begingroup$ $\theta$ is a random variable. $\endgroup$ – Tochoka Dec 7 '18 at 20:57
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    $\begingroup$ Yeah I know, and the solution does not use this. $\endgroup$ – Did Dec 7 '18 at 21:13

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