- $(\Omega, \mathcal{F}, \mathbb{P})$ probability space
- $(E, \mathcal{B}(E))$ a measurable space where $B(E)$ is the Borel sigma algebra and $E$ is a Banach space
- $X:\Omega\to E$ random variable with distribution $\mathbb{P}^X$ on $(E, \mathcal{B}(E))$
I know that one can write the probability of an event as the expectation of an indicator function $$ \mathbb{E}[1_A] = \int_{\Omega} 1_A(\omega) \mathbb{P}(d\omega) = \int_A d\mathbb{P} = \mathbb{P}(A). $$
Can I also write $\mathbb{P}^X(A) = \mathbb{E}[1_A X]$ for $A\in\mathcal{B}(E)$?
I have tried directly $$ \begin{align} \mathbb{P}^X(A) &= \int_A \mathbb{P}^X(de) \\ &= \int_E 1_A(e) \mathbb{P}^X(de) \end{align} $$ but I can't use the change of variables formula here, even if it is very suspitious. The random variable takes inputs in $\Omega$ not in $E$, I would be tempted to write $$ \int_\Omega 1_A(X(\omega)) \mathbb{P}(d\omega) $$
$$\mathbb P^X \left[A\right] = \mathbb P\left[X\in A\right] = \mathbb E\left[\mathbf 1_{X\in A}\right]$$
The $A$ is Borel and $\left\{X\in A\right\} \in \mathcal B(E)$