Showing a probability space I have recently been exposed to probability measure theory and found an exercise in my textbook which I am not able to solve.
I have to show that ($\mathbb R, \mathscr B(\mathbb R), \mu_X $) is a probability space, where $\mathscr B(\mathbb R)$ represents the Borel algebra of $\mathbb R$ and $\mu_X$ is defined as a function $\mu_X(B) = \mathbb P(X^{-1}(B)), \forall B\in \mathscr B (\mathbb R) $.
It gives a hint of using the fact that $\mathbb P$ is a probability measure and the definition of a random variable, which I would define as a function $X: \Omega \rightarrow\mathbb R $.
Can anyone help me with this and show me how to go about proving this? Thanks!

In order to show that $\mu_X$ is a probability measure, we first have to check that it is indeed a measure. Since the inverse image by $X$ of the empty set is the empty set, it suffices to show the $\sigma$-additivity. Let $\left(B_i\right)_{i\in\mathbb N}$ be a collection of pairwise disjoint Borel sets. Noticing that $\left(X^{-1}B_i\right)_{i\geqslant 1}$ is also pairwise disjoint and that $\mathbb P$ is a measure, we get that 
$$
\mu_X\left(\bigcup_{i\in\mathbb N}B_i\right)=\mathbb P\left(\bigcup_{i\in\mathbb N}X^{-1}B_i\right)=\sum_{i\in\mathbb N}\mathbb P\left(X^{-1}B_i\right)=\sum_{i\in\mathbb N}\mu_X\left(B_i\right).
$$
That $\mu_X(\mathbb R)=1$ follows from $X^{-1}(\mathbb R)=\Omega$.
