# Expected function evaluation of random variable w.r.t. different distribution

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