We show that if (X, μ, T) is a probability measure-preserving dynamical system, and P is a countable partition of (X, μ), then the limit ₍, ₊ E 1k ₉ = ₀^k - 1 f Tʲ ₈ = ₀^n - 1 T^-i P exists almost surely for all f Lᵖ (μ), p > 1. We prove this as a corollary of a geometric result: that if (X, μ) is a metric measure space on which the Hardy-Littlewood maximal inequality holds, then the limit ₑ ₀, ₊ μ (B (x, r) ) ^-1 ₁ (ₗ, ₑ) 1k ₉ = ₀^k - 1 f Tʲ d μ exists almost surely.
Aidan Young (Thu,) studied this question.