A probability space consists of the following data:
The sample space , which is the set of possible outcomes (of an experiment.
The event algebra , where each event consists of a set of outcomes in , and the collection of events constitutes a -algebra – it is closed under countable sequences of union, intersection and complement operations (and hence set differences). Implied here is that the empty set and whole sample space are events in .
A measure function , which assigns a probability to each event in . must be additive on countable disjoint unions, and must assign 1 to the whole sample space .
A random variable is a function from the sample space S into a range space , which is measurable, which means: there is a -algebra of subsets of , and the inverse image of every such subset under the function is an event in .