Hi I am struggling to understand this notation for conditional expectation:
(Say $X_{t}$ is a process that takes values in $\mathbb{R}$) then $$E[f(X_{t})|X_{0}=x]=\int_{\mathbb{R}}f(y)p_{t}(x,dy)$$ Where $p_{t}(x,dy)$ is the transition probability function for the process $X_{t}$.
I understand (since $x$ is fixed) $p_{t}(x,dy)$ is a probability measure, but I have not really seen integrals described using measures before...
The right side is just a complicated definition for some function of $x$. And the notation $E[Y \mid X=x] = f(x)$ simply means that $E[Y \mid X] = f(X)$, an (almost sure) equality of random variables. (Note that the function $f$ is not necessarily unique, so this needs to be taken with a small grain of salt.)