Orlicz spaces which areL p -spaces

Abstract

Let Ф denote a finite-valued Orlicz function vanishing only at zero and consider the Orlicz spaceL Ф over a non-atomic, σ-finite and infinite measure space with the norm ∥ · ∥ being either the Luxemburg norm or (in the case where $$\mathop {\lim }\limits_{u \to 0} \Phi (u)/(u) = 0$$ and $$\mathop {\lim }\limits_{u \to + \infty } \Phi (u)/(u) = + \infty $$ the Orlicz norm. Ifp ∈ [1, + ∞) and $$||1_A + 1_B ||^P = ||1_A ||^P + ||1_B ||^P $$ for all disjoint setsA andB of positive and finite measure, thenL Ф is the Lebesgue spaceL p .