Normed Space Of Measurable Functions With Some Of Their Properties

Abstract

Let L ⁰(Ω, F, μ) be the space of measurable functions defined on measure space (Ω, F, μ), where we consider any two functions in which are equal almost everywhere (a. e). Then L ⁰(Ω, F, μ) is complete metric space with respect to metric functions defined by \(d(f,g)={\int }_{{\rm{\Omega }}}\frac{|f-g|}{1+|f-g|}d\mu \) for all f, g ∈ L ⁰(Ω, F, μ). This paper includes two main parts, the first part we prove this space L ⁰(Ω, F, μ) in general is not a normed space, and second we prove norm on L ⁰(Ω, F, μ) achieved if and only if she was Ω is the finite union of disjoint atom.