Real-valued function
Appearances in measure theory

A measure on a set is a non-negative real-valued functional on a σ-algebra of subsets. L spaces on sets with a measure are defined from aforementioned real-valued measurable functions, although they are actually quotient spaces. More precisely, whereas a function satisfying an appropriate summability condition defines an element of L space, in the opposite direction for any f ∈ L (X) and x ∈ X which is not an atom, the value f (x) is undefined. Though, real-valued L spaces still have some of the structure described above in § Algebraic structure. Each of L spaces is a vector space and have a partial order, and there exists a pointwise multiplication of "functions" which changes p, namely

For example, pointwise product of two L functions belongs to L.
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