Let $E$ be an event and $Y$ a random variable. What exactly is meant by $\mathrm E[\mathbf 1_E \mathbf 1_{Y\in B}]$? I have two guesses, the first is that $\mathbf 1_{Y\in B}$ is an indicator random variable and the second is that $\mathbf 1_{Y\in B}$ is really a composition of an indicator function and a random variable (hence a random variable):
\begin{align*} \mathrm E[\mathbf 1_E \mathbf 1_{Y\in B}] & = \int_{\Omega} \mathbf 1_E(\omega) \mathbf 1_{Y^{-1}(B)}(\omega)\, \mathrm dP(\omega) \\ \mathrm E[\mathbf 1_E \mathbf 1_{Y\in B}] & = \int_{\Omega} \mathbf 1_E(\omega) (\mathbf 1_B \circ Y)(\omega)\, \mathrm dP(\omega) \end{align*}
I think it's the first line, since then we could write $$ \mathbf 1_E(\omega) \mathbf 1_{Y^{-1}(B)}(\omega) = \mathbf 1_{E\, \cap\, Y^{-1}(B)}(\omega) = \begin{cases} 1, \omega \in E\, \cap\, Y^{-1}(B), \\ 0, \omega \notin E\, \cap\, Y^{-1}(B). \end{cases} $$ I'm not sure how we could write this for the second line, or even if it would make sense to try.
Also, assuming the first line is correct I don't immediately see how it would be possible to write this using the distribution measure of $Y$, $Y_*[P]$. For example, if we only had $\mathrm E[\mathbf 1_{Y\in B}]$, I would like to think of this as an indicator function and write $$ \mathrm E[\mathbf 1_{Y\in B}] = \int_\Omega (\mathbf 1_{B} \circ Y)(\omega)\, \mathrm dP(\omega) = \int_{Y(\Omega)} \mathbf 1_B(y)\, \mathrm d(Y_*[P])(y) $$
On the other hand I guess we can use the definition of regular conditional probability to write the integral w.r.t. the distribution of $Y$: $$ \mathrm E[\mathbf 1_E \mathbf 1_{Y\in B}] = P(E \cap \{Y \in B\}) = \int_B P^Y(E \mid y)\, \mathrm d(Y_*[P])(y) $$ where the second equality is the definition of the regular conditional probability of $P$ given $Y$, $P^Y$. Would this be the proper thing to do?
The notation $\mathbf{1}_E$ denotes an indicator variable which is equal to 1 if the event $E$ holds, and to 0 otherwise. In your case, there are two events, $E$ and $Y \in B$. The expression $\mathbf{!}_E \mathbf{!}_{Y \in B}$ is simply the product of the two indicator variables. All in all, it equals the probability that both $E$ happens and $Y \in B$.