Random Variable Laws

Why

We name the image measures of real-valued random variables.

Definition

The law of a random variable is the image measure of the probability measure under the random variable.

For example, if the random variable is real-valued we use the topological sigma algebra of the real numbers and the law is the image measure on $\R $ induced by the probability measure.

Notation

Let $(X, \mathcal{A} )$ and $(Y, \mathcal{B} )$ be two measurable spaces. Let $f: X \to Y$ be a random variable. Let $\mu : \mathcal{A} \to \nneri$ be a probability measure. We denote the law of $f$ by $\rvlaw{\mu }{f}$.