It is known that if a
derivation basis ℬ possesses Vitali-like covering properties, with covering families
having arbitrarily small L(p)(μ)-overlap, where 1 ≦ p < +∞ and μ is a σ-finite
measure in an abstract measure space, then ℬ derives the μ-integrals of all functions
f ∈ L(q)(μ) where p−1+ q−1= 1 if p > 1; q = +∞ if p = 1. The converse is well
known for the case q = +∞, p = 1, and a partial converse is known for the case
p > 1, if ℬ is a [U,δ]-basis. The present paper offers a converse for p > 1 under
general hypotheses and, simultaneously, removes the necessity that ℬ be a
[U,δ]-basis.
