An ergodic Lebesgue differentiation theorem

2025/06/26 by Young, Aidan
#Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.2506.21421

Abstract

We show that if (X, μ, T) is a probability measure-preserving dynamical system, and \mathscrP is a countable partition of (X, μ), then the limit lim_{n, k → ∞} 𝔼 [ (1)/(k) ∑_{j = 0}^{k - 1} f ∘ T^j | \bigvee_{i = 0}^{n - 1} T^-i \mathscrP ] exists almost surely for all f ∈ L^p(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 lim_{r \searrow 0, k → ∞} μ(B(x, r))^-1 ∫_{B(x, r)} (1)/(k) ∑_{j = 0}^{k - 1} f ∘ T^j d μ exists almost surely.

Citations

• Spatial-Temporal Differentiation Theorems

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