# Measurable flow

*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.

**How to Cite This Entry:**

Measurable flow.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Measurable_flow&oldid=11332