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 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.
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