Recurrence in mean for group actions

In this paper, the mean values of the recurrence are computed for general group actions. Let (X, d) be a metric space with a finite measure mu and G be a countable group acting on (X, d). Let F-1, F2, ... be a sequence of subsets of G with vertical bar F-n vertical bar -> infinity and put E-n = F-n(-1) F-n. If the Hausdorff measure H-h is finite on X and mu is T-invariant. We assume that mu and H-h are concordant. Then the function C(x) := lim inf (n ->infinity) (vertical bar Fn-1 vertical bar . min(g is an element of En\{e}) h(d(x, gx))) is mu-integrable and for any mu-measurable set B, we have integral(B) C(x)d mu <= H-h (B). If moreover, H-h(X)= 0, then integral(B) C(x)d mu = 0 without the concordance condition for the measure mu and H-h.