Norms supporting the Lebesgue differentiation theorem

Abstrakt

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.

Klíčová slova

• Lebesgue differentiation theorem
• rearrangement-invariant spaces
• Lorentz spaces
• Orlicz spaces
• Marcinkiewicz spaces