Generalization of a theorem by Hardy,Littlewood, and Pólya

Let τ be a non-expanding map which admits a σ-finite, absolutely continuous invariant measure μ. The measure $\text{μ¦}{}_{\mathrm{E}}$ is approximated as closely as desired by a measure μ_n invariant under the transformation τ_n satisfying $\text{inf}{}_{\mathrm{x}}\text{¦τ′}{}_{\mathrm{n}}\text{(x)¦ > 1}$. The orbit {τ_n^j(x)}_{j = 0}^∞ “exhibits” $\text{μ¦}{}_{\mathrm{E}}$ much more quickly than does the orbit {τ^j(x)}_{j = 0}^∞.