Thin subspaces of L ¹(λ)

Abstract

Let (S,Σ, λ) be a finite positive measure space. In their extension of Liapunov's theorem to infinite dimensions, Kingman and Robertson (in 1968) defined a subset N of the real L ¹(λ) to be thin if for every A ∈ Σ with λ(A) > 0 there exists a φ ₀ ∈ L ^∞(λ) with φ ⁰ ≠O, supported on A that “annihilates N”. We prove that the φ ⁰ above can be chosen to have values ±1 a.e. on A, i.e. a “sign” in terms of a (2009) paper by Wulbert. A result of his follows from the known fact that when λ is non-atomic every finite dimensional subspaces of L ¹(λ) is thin. Further, in this case when S is compact Hausdorff and A compact, we show that φ ₀ may be chosen in C(A). Next we have a “splitting” of Liapunov's theorem in any dimension into “convex ranges” and “weakly compact ranges”, the latter for non-atomic measures. This approach is related to a 1978 result by K.M. Garg and the author. We observe the closed linear span of the Rademacher functions to be thin in L ¹[0, 1].