Markov property for Brownian motion; meaning of $P[B_t\in A|\mathscr{F}_S^B](\omega)$?

Question

I am reading these notes on Brownian motion. On page 22 it shows that:

$$P[B_t\in A|\mathscr{F}_S^B](\omega)=\int_A (2\pi(t-s))^{-d/2}\exp\left(-\frac{|y-B_s(\omega)|^2}{2(t-s)}\right)dy\quad \text{P-almost surley}$$ for a Brownian motion $(B_t)_{t\ge 0}$ in $\Bbb{R}^d$ over $(\Omega,\mathscr{F},P)$ with $A \subseteq \Bbb{R}^d$. I am confused here about the meaning of: $P[B_t\in A|\mathscr{F}_S^B](\omega)$. I see two problems with it:

1. $B_t$ is a random variable and as such cannot be in $A$ or for that matter $\Bbb{R}^d$ unless its argument $B$ is explicit.
2. The action of $P$ must be on elements of $\mathscr{F}$ (this is how it is defined). In this case it is not (even if $B_t$ where taken as the random variables).

My question is therefore what does $P[B_t\in A|\mathscr{F}_S^B](\omega)$ mean? and in what space does everything live?

Answer 1

If $X:\Omega\to\mathbb R$ is a random variable, and $A\subset\mathbb R$ is a Borel set, then $[X\in A]$ is shorthand for the event $\{\omega\in\Omega:X(\omega)\in A\}$. This is extremely widely used, making it quite surprising that you have studied both Brownian motion and conditional expectation without encountering it.