A random variable is a function that transform a probability space to another space is built up on real number
i.e ($\Omega,\mathcal F,P)\to(\Bbb R,\mathcal B,P_X$)
and a random variable is call measurable if $\forall B \in \mathcal B$ $X^{-1}(B)\in\mathcal F$
How do we guarantee that$P(A)$ $\forall A\in \mathcal F $ is equal to $P_X(B)\forall B\in \mathcal B$
or How we show that $\int _AdP=\int _BdP_X$ where $A=$ $X^{-1}(B)$ $\forall B\in \mathcal B$
I know this maybe intuitive but I want have a strict mathematics proof to show this two things is equal
That's because the construction of the probability measure $P^{X}$.
You can see that $\forall B\in \mathcal{B}, \hspace{0.3 cm}$ $P^{X}(B):=P(X^{-1}(B))$ is indeed a probability measure in the space $(\mathbb{R},\mathcal{B})$. (Check that it satisfies the axioms of probability)
Therefore, $ \forall B\in \mathcal{B} \hspace{0.2 cm}$ if we define $A=X^{-1}(B)\in \mathcal{F}$
we have that $P^{X}(B)=P(X^{-1}(B))=P(A)$