Volume of spheres in doubling metric measured spaces and in groups of polynomial growth

Let G be a compactly generated locally compact group and let U be a compact generating set. We prove that if G has polynomial growth, then (U)n∈N is a Følner sequence and we give a polynomial estimate of the rate of decay of μ(Un+1 r Un) μ(Un) . Our proof uses only two ingredients: the doubling property and a weak geodesic property that we call Property (M). As a matter of fact, the result remains true in a wide class of doubling metric measured spaces including manifolds and graphs. As an application, we obtain a Lp-pointwise ergodic theorem (1 ≤ p < ∞) for the balls averages, which holds for any compactly generated locally compact group G of polynomial growth.