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.