A generalization of a lemma of Kac

Denote by t (k,.) : A -> N the k-th return function for a set A with positive measure and a transformation T : Omega -> Omega that preserves mu. Kac's lemma asserts that integral(A) iota (1,.) = mu(Omega) - mu(E *). Where E* is the set of points that never enter A. We generalize this formula for arbitrary time of return: integral(A) iota (k,.) = k(mu(Omega) - mu(E*)). We also prove that the distributions of iota (1,.) and t (k,.)- iota(k - 1,.) are equal for arbitrary k.