Probability Distribution is Probability Measure

Theorem
Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.

Let $\struct {S, \Sigma'}$ be a measurable space.

Let $X$ be a random variable on $\struct {\Omega, \Sigma, \Pr}$ taking values in $\struct {S, \Sigma'}$.

Let $P_X$ be the probability distribution of $X$.

Then:


 * $P_X$ is a probability measure on $\struct {S, \Sigma'}$.

Proof
From the definition of probability distribution, we have:


 * $P_X = X_* \Pr$

where $X_* \Pr$ is the pushforward $X_* \Pr$ of $\Pr$, under $X$, on $\Sigma'$.

From Pushforward Measure is Measure, we have:


 * $P_X$ is a measure.

We then have:

so:


 * $P_X$ is a probability measure.