Measurable flow

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in a measure space

A family ( runs over the set of real numbers ) of automorphisms of the space such that: 1) for all , ; and 2) the mapping taking to is measurable (a measure is introduced on as the direct product of the measure in and the Lebesgue measure in ). "Automorphisms" here are to be understood in the strict sense of the word (and not modulo 0), that is, the must be bijections carrying measurable sets to measurable sets of the same measure. In using automorphisms modulo 0, it turns out to be expedient to replace condition 2) by a condition of a different character, which leads to the concept of a continuous flow. Measurable flows are used in ergodic theory.

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Measurable flow. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by D.V. Anosov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article