  A version of the Lebesgue differentiation theorem is offered, where the $L^p$
norm is replaced with any rearrangement-invariant norm. Necessary and
sufficient conditions for a norm of this kind to support the Lebesgue
differentiation theorem are established. In particular, Lorentz, Orlicz and
other customary norms for which Lebesgue's theorem holds are characterized.

Functional Analysis