Let's say I have a stochastic process, $S_t$ which is measurable and adapted to a natural Brownian filtration. Let's assume it is a 1-dimensional process in the reals. I want I write the expected value of a function of this process as follows $$\mathbb{E}[f(S_T)]$$
I want to write this in the integral form, my question here is the correct way to write this? Are we integrating w.r.t. the Lebesgue measure?
$$\mathbb{E}[f(S_T)] = \int f(S_T) d\mu$$
What is the proper way to write this expected value in integral form?
The expected value would be written as $$E[f(S_T)] = \int_\Omega f(S_T) d \mathbb{P}$$ where $\mathbb{P}$ is the ambient probability measure. If you know the density of $S_T$, call it $g$, then you can evaluate this as an integral with respect to the Lebesgue measure $$E[f(S_T)] = \int_{\mathbb{R}} f(x) g(x) d x$$