A probability space consists of the following data:

The sample space$S$ , which is the set of possible outcomes (of an experiment) experiment.

The event algebra$A$, where each event consists of a set of outcomes in $S$, and the collection of events constitutes a $\sigma$ -algebra – it is closed under countable sequences of union, intersection and complement operations (and also hence set differences). Implied here is the fact that the empty set and whole sample space are events in$A$.

A measure function$P$, which assigns a probability to each event in $A$ . P must be additive on countable disjoint unions, and must assign 1 to the whole sample space$P$ must be additive on countable disjoint unions, and must assign 1 to the whole sample space $S$.

A random variable is a function $X$ from the sample space S into a range space $V$, which is measurable, which means: there is a $\sigma$-algebra of subsets of $V$, and the inverse image of every such subset under the function $X$ is an event in $A$.