Equivalence of Expectation and Probability

Question

In a book that I'm reading at the moment, the author claims that given a sample space $\Theta$ and a random variable $\theta \sim G$ on $\Theta$ for some random measure $G$, that for measurable $A\subset\Theta$ we have $\mathbb{P}(\theta \in A) = \mathbb{E}[G(A)]$. But I don't know why this is true. In particular isn't $\mathbb{P}(\theta \in A) = \mathbb{P}\circ\theta^{-1}(A) = G(A)$?

Answer 1

Saying that a random variable is distributed according to a measure $G$ (Symbol $\theta \sim G$) means by definition that

$$ P( \theta \in A) = G(A) $$ holds for all measurable $A$ and since $G(A)$ is a constant, we have of course $$ \mathbb{E}[G(A)] =G(A)$$ and thus $$ P( \theta \in A) = \mathbb{E}[G(A)] .$$

Another maybe more relevant thing to note is the following relation: $$ \mathbb{P}[ \theta \in A ]= \int_E 1_{\{\theta^{-1}(A)\}}(x) dP = \mathbb{E}[1_{\{\theta^{-1}(A)\}}].$$

BTW: In which book did you read this?