Suppose I have two continuous random variables on the same domain, $\xi \sim \mathbb{P}, \xi' \sim \mathbb{Q}, \in \Xi$ and joint probability $(\xi, \xi') \sim \Pi \in \Xi^2$ . Now I would like to know the expectation of a function $f$ of $\xi$ on the distribution $\mathbb{Q}$:= $\quad \quad \mathbb{E}^\mathbb{Q}[f(\xi)]$

I believe the right integral would be:

$\mathbb{E}^\mathbb{Q}[f(\xi)] = \int_{\Xi^2}f(\xi)\mathbb{}\Pi(d\xi,d\xi')$

However, I rather think it should be the following integral: $\mathbb{E}^\mathbb{Q}[f(\xi')] = \int_{\Xi}f(\xi')\mathbb{Q}(d\xi')$


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