Dichotomies for spaces

Assume that ( X , Σ , μ ) is a measure space and p 1 , … , p n , r > 0 . We prove that { ( f 1 , … , f n ) ∈ L p 1 × ⋯ × L p n : f 1 ⋯ f n ∈ L r } is either L p 1 × ⋯ × L p n or a σ-porous subset of L p 1 × ⋯ × L p n . This dichotomy depends on properties of μ and the sign of the number 1 r − 1 p 1 − ⋯ − 1 p n .