Is the distribution of a random variable always a probability measure? Is the distribution of a random variable always a probability measure? Let $\mathbb{P}X^{-1}$ denote the distribution of the random variable $X$.
For example, can I denote the distribution of a normal random variable as a probability measure?
 A: Yes, by definition, the distribution of a real random variable $X$, which is a measurable function on $(\Omega,\mathcal{F},\mathbb{P})$, is the probability measure $\mathbb{P}_X$ defined on the measurable space $(\mathbb{R},B(\mathbb{R}))$ (with $B(\mathbb{R})$ the Borel sigma-algebra) by $\mathbb{P}_X(A)=\mathbb{P}(X\in A)$. So we can write that $\mathbb{P}_X=\mathbb{P}\circ X^{-1}$.
It is easy to show that it defines a probability measure on $(\mathbb{R},B(\mathbb{R}))$. It is actually a special case of a pushforward measure (see https://en.wikipedia.org/wiki/Pushforward_measure ).
Note that if $X$ is not real but $X$ has its values on a more general measurable space $(S,\mathcal{S})$, the definition is the same, and the distribution of $X$ will be a probability measure defined on $\mathcal{S}$.
A: It is definitely a probability measure on the reals, by definition: $P_X([a,b])=P(a\leq X(\omega)\leq b).$
For a standard normal variable the probability of an interval $[a,b]$ is the integral of the density function (the Gauss curve) between the end points, or equivalently, the difference in value of the error function between those two arguments.
