Questions about maps from a probability space to a measure space which are measurable.

A random variable $X: \Omega \to E$ is a measurable function from a set of possible outcomes $\Omega$ to a measurable space $E$. The technical axiomatic definition requires $\Omega$ to be a sample space of a probability triple. Usually $X$ is real-valued.

The probability that $X$ takes on a value in a measurable set $S \subseteq E$ is written as :

$$P(X \in S) = P(\{ \omega \in \Omega|X(\omega) \in S\})$$

where $P$ is the probability measure equipped with $\Omega$.