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The number assigned by a probability measure to an event.
A probability of an event in a probability space is the number assigned to an event .
Because events are measurable sets and is a probability measure, event probabilities satisfy the usual axioms (nonnegativity, countable additivity on disjoint events, and normalization).
A probability space is a triple where is a set, is a sigma-algebra on , and is a probability measure on .
Elements of are the events whose probabilities are evaluated by , and a random variable is a measurable function defined on .
A measurable set in a measurable space is a subset with .
Measurable sets are precisely the subsets to which a measure assigns a value, and they determine measurable functions via preimages. The indicator function of a measurable set is a basic example of a measurable function.
A probability measure is a function defined on a sigma-algebra of subsets of a set such that , for every pairwise disjoint sequence , and .
A probability measure is a special case of a measure and is the key ingredient in a probability space; it assigns probabilities to events (measurable sets).