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In mathematics, a probability measure is a real-valued function defined on a set of events in a probability space that satisfies measure properties such as ''countable additivity''.〔''An introduction to measure-theoretic probability'' by George G. Roussas 2004 ISBN 0-12-599022-7 (page 47 )〕 The difference between a probability measure and the more general notion of measure (which includes concepts like area or volume) is that a probability measure must assign value 1 to the entire probability space. Intuitively, the additivity property says that the probability assigned to the union of two disjoint events by the measure should be the sum of the probabilities of the events, e.g. the value assigned to "1 or 2" in a throw of a die should be the sum of the values assigned to "1" and "2". Probability measures have applications in diverse fields, from physics to finance and biology. ==Definition== The requirements for a function to be a probability measure on a probability space are that: : * must return results in the unit interval (1 ), returning 0 for the empty set and 1 for the entire space. : * must satisfy the ''countable additivity'' property that for all countable collections of pairwise disjoint sets: :: For example, given three elements 1, 2 and 3 with probabilities 1/4, 1/4 and 1/2, the value assigned to is 1/4 + 1/2 = 3/4, as in the diagram on the right. The conditional probability based on the intersection of events defined as: : satisfies the probability measure requirements so long as is not zero.〔''Probability, Random Processes, and Ergodic Properties'' by Robert M. Gray 2009 ISBN 1-4419-1089-1 (page 163 )〕 Probability measures are distinct from the more general notion of fuzzy measures in which there is no requirement that the fuzzy values sum up to 1, and the additive property is replaced by an order relation based on set inclusion. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Probability measure」の詳細全文を読む スポンサード リンク
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