What does it mean $\mathbb E[\Phi(x,Y)]|_{x=X}$ in the book of R. Schilling (Brownian motion)?

Question

In a book, there are the following notations that I don't understand.

know conditional expectation, but here I don't get what $\mathbb E[\Phi(x,Y)]|_{x=X}$ means. Is it $\mathbb E[\Phi(X,Y)\mid X]$ or $\mathbb E[\Phi(X,Y)\mid X=x]$ ?

---

Answer 1

1. It means that for every Borel subset $B$ of $\mathbb R^d$, $X^{-1}(B):=\{\omega,X(\omega)\in B\}$ belongs to $\mathcal X$.

2. We can define a function $f\colon\mathbb R^d\to\mathbb R$ by $f(x)=\mathbb E\left[\Phi\left(x,Y\right)\right]$ and the notation $\mathbb E[\Phi(x,Y)]|_{x=X}$ means that for each fixed $\omega$, we evaluate $f$ at $X(\omega)$.

Answer 2

I'm not a huge fan of these notation, but it looks that if $f(x)=\mathbb E[\Phi(X,Y)\mid X=x]$ then $$\mathbb E[\Phi(x,Y)]|_{x=X}=f(X)=:\mathbb E[\Phi(X,Y)\mid X].$$