Proof of existence of probability distribution: Why $P((-\infty,a])=P^{*}(X^{-1}((-\infty,a]))$?

Question

So I'm looking at the proof that for a given cdf we can always find a probability distribution that corresponds to it.

In the proof they start out with the probability space $(\Omega,\mathcal{B}(0,1),P^*)$ where $P^*$ is the uniform distribution on $(0,1)$.

They construct the random variable $X$ as $X(t)=\text{min}\{x:F(x)\geq t\}$ and show that

$\forall a\in\mathbb{R}:X^{-1}((-\infty,a])=(-\infty,F(a)]\in\mathcal{B}(0,1)$

Now to show that $P((-\infty,a])=F(a)$ there is this one step wich I don't really understand:

$P((-\infty,a])=P^*(X^{-1}((-\infty,a]))$

On a side note, I have no knowledge of measure theory since at my university it wasn't a prequisite for taking this course on probability.

Answer 1

So we have a random variable $X: \Omega \to \mathbb{R}$ (with $P^*$ been probability measure on $\Omega$). By definition, it's probability distribution is a measure $P$ on $\mathbb{R}$ such that $P(A) = P^*(\{t | X(t) \in A\}) = P^*(X^{-1}(A))$.