NEW CHARACTERIZATIONS OF SOME L p -SPACES

For a complete measure space (X, Σ ,µ) , we give conditions which force L p (X, µ), for 1 ≤ p 0 and for every E ⊂ A with E ∈ Σ, either µ(E) = 0o rµ(E) = µ(A). A measurable subset E with µ(E) > 0i snonatomic if it does not contain any atom. We say that two atoms A1 and A2 are distinct if µ(A1 ∩ A2) = 0. We say that two atoms A1 and A2 are indistinguishable if µ(A1 ∩ A2) = µ(A1) = µ(A2). A measurable space (X, Σ ,µ) is said to be atomic if every measurable set of positive measure contains an atom. For more information on measurable spaces and related topics we refer to (1, 2, 4). We collect some interesting and useful properties of atomic and nonatomic sets in the following proposition.