Existence of law

Question

How do you prove that it exists a probability space $(A,\mathcal{A}, \mathcal{P})$ and a random variable $X$ such that $$ \mathcal{P}_{X} = f d\mathcal{L} $$ with $f(x) = \exp\left(-\frac{x^{2}}{2}\right)$

My question is not specific to the Gaussian law, I can ask the same thing for any law.

For the answer a ref will be enough, thanks and regards.

Answer 1

Take space $(A,\mathcal A,\mathcal P)=(\mathbb R,\mathcal B(\mathbb R),\mathcal P)$ where $\mathcal P$ is prescribed by $B\mapsto\int_Bfd\mathcal L$.

Then prescribe random variable $X:A\to\mathbb R$ by $a\mapsto a$ (i.e. the identity function).

Then $\mathcal P_X=\mathcal P=fd\mathcal L$.