Kac's lemma

In ergodic theory, Kac's lemma, demonstrated by mathematician Mark Kac in 1947,^[1] is a lemma stating that in a measure space the orbit of almost all the points contained in a set ${\displaystyle A}$ of such space, whose measure is ${\displaystyle \mu (A)}$, return to ${\displaystyle A}$ within an average time inversely proportional to ${\displaystyle \mu (A)}$.^[2]

The lemma extends what is stated by Poincaré recurrence theorem, in which it is shown that the points return in ${\displaystyle A}$ infinite times.^[3]