# Measurable flow

in a measure space $(M,\mu)$
A family $\{T^t\}$ ($t$ runs over the set of real numbers $\mathbf R$) of automorphisms of the space such that: 1) $T^t(T^s(x))=T^{t+s}(x)$ for all $t,s\in\mathbf R$, $x\in M$; and 2) the mapping $M\times\mathbf R\to M$ taking $(x,t)$ to $T^tx$ is measurable (a measure is introduced on $M\times\mathbf R$ as the direct product of the measure $\mu$ in $M$ and the Lebesgue measure in $\mathbf R$). "Automorphisms" here are to be understood in the strict sense of the word (and not modulo 0), that is, the $T^t$ must be bijections $M\to M$ 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.