Measurable flow
in a measure space
2020 Mathematics Subject Classification: Primary: 37A10 [MSN][ZBL]
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.
Measurable flow. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Measurable_flow&oldid=21665