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A probability space consists of the following data:
The sample space $S$, which is the set of possible outcomes (of an 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 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 $S$.