Definition

measure induced by random element

If $(\Omega, \mathcal{B}, \mathbb{P})$ is a probability space and $X: (\Omega, \mathcal{B}) \to (\Omega ', \mathcal{B} ')$ is measurable, then the function $\mathbb{P} \circ X^{-1}: \mathcal{B} ' \to [0, 1]$ is a probability measure on $(\Omega ', \mathcal{B} ')$ called the probability measure induced by $X$. (For $A ' \sub \Omega '$ we define $[X \in A '] = X^{-1}(A ')$.)

If $X$ is a random variable then the induced measure coincides with that determined by its distribution function.