Does every density have random variable.

Question

I am studying probability, everywhere i read, the following is stated as a fact: if $f$ is measurable and positive, and $$\int_{\mathbb{R}}f(x)dx = 1$$ then $f$ is density of some random variable. Is it really that obvious? I am unable to construct a random variable out of it. I'd be greatful for outline of a proof.

Answer 1

You are asking a technical question whose answer is quite trivial knowing the right knowledge. Here is it: a random variable is, by definition, a measurable function $X$ from some probability space $(\mathrm{X}, \mathscr{X}, \mathbf{P})$ into the real numbers with Borel sets. If $f$ is a density, then you can define $\mathbf{P}(\mathrm{A}) = \int\limits_{\mathrm{A}} f$ and show this is a probability measure on the Borel sets of $\mathbf{R}.$ On the probability space $(\mathbf{R}, \mathscr{B}, \mathbf{P})$ define the random variable $X(t) = t,$ in simple English, $X$ is the identity. Then, it follows trivially that $X$ is a random variable whose distribution is $\mathbf{P}$ and, by definition of $\mathbf{P},$ the density of $X$ is $f.$