On differentiation of integrals in Lebesgue spaces

We study the problem of differentiation of integrals for certain bases in the infinite-dimensional torus Tω . In particular, for every p₀ ∈ [1,∞) , we construct a basis B which differentiates Lᵖ(Tω) if and only if p ≥ p₀ , thus reproving classical theorems of Hayes in R . The main novelty is that our B is a Busemann–Feller basis consisting of rectangles (of arbitrarily large dimensions) with sides parallel to the coordinate axes. Our construction gives us the opportunity to classify all possible ranges of differentiation for general complete spaces. Namely, let B be a basis in a metric measure space X . If X is complete, then the set \{ p ∈ [1,∞] : B differentiates Lᵖ(X) \} takes one of the six forms ∅ , \{∞\} , [p₀,∞] , (p₀,∞] , [p₀,∞) , (p₀,∞) for some p₀ ∈ [1,∞) . Conversely, for every p₀ ∈ [1,∞) and each of the six cases above, we construct a complete space X and a basis B illustrating the corresponding range of differentiation.