In probability theory, a probability space is a measure space whose measure is a probability distribution: its integral is (e.g. Dembo 12, 1.1).
One thinks of the elements as possible configurations of a system subject to randomness, hence of as a space of “possible worlds” in the idealized situation under consideration, and for any subset one thinks of as the probability that the system is found in a configuration which lies in .
Accordingly, a measurable function on a probability space has the interpretation of a random variable. Its integral is its expectation value.
The modern formal concept originates around
Surveys and lecture notes include
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