Year: 1968
Type: article
Abstract: For a space (S, Σ, μ), μ a positive finitely additive set function on a field Σ of subsets of the set S, Z# P (S, -£, /*) is usually not complete.However, if we consider the completion L P (S, Σ, μ) of L Pf we may ask which of the properties of L p known for the countably additive case, are true in general.In this paper it is shown that for every (S, Σ, μ) there is a (countably additive) measure space (S', Σ f , μ') and a natural injection j from S into S r which induces isometric isomorphisms y* from L P (S, Σ, μ) onto L P (S', Σ f , //).j* also preserves order, and other structures on L p .This result shows, roughly, that any theorem valid for L p over a measure space, applies also to L p over a finitely additive measure.Thus L p and L q are dual (1 < p < + oo, Ijp + Ijq -1), Li is weakly complete, and so forth.Let S be a set, Σ a field of subsets of S, and μ a finitely additive extended real-valued set function on Σ.We call (S, 2\ μ) a triple.If μ is positive or bounded, we call (S, Σ, μ) a positive or bounded triple, respectively.Let / be a /^-simple function on S. We define the L^-norm of /,
Source: Pacific Journal of Mathematics
Author: Charles Fefferman
Language: English
Topic: Advanced Banach Space Theory
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: bronze
OpenAlex ID: W2093304082