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Measurable functions with finite L^p norm, identified up to equality almost everywhere.
For a measure space and , the space is
where is or , is the norm, and means that almost everywhere.
The quotient makes the norm a genuine norm rather than a seminorm. The cases and correspond to functions and functions, respectively.
A measure space is a triple where is a measurable space and is a measure on .
Measure spaces provide the basic setting for integration, convergence theorems, and statements that hold almost everywhere rather than pointwise.
A norm on a measure space (for ) assigns to a measurable function the value
whenever the Lebesgue integral of the nonnegative function is finite; here uses the absolute value. For one defines
using the essential supremum.
If and are equal almost everywhere, then , so the norm is naturally a norm on the corresponding space.
A property is said to hold almost everywhere (with respect to a measure on ) if there exists a null set such that holds for all .
This notion depends on the underlying measure space and is weaker than pointwise validity: changing a function on a null set does not affect almost-everywhere statements. It is also closely tied to the symmetric difference of sets.
An function on a measure space is an equivalence class of measurable functions such that
where two functions are identified if they satisfy a.e. equality.
Choosing any representative of the class, the quantity is its Lp norm with , and the set of all such classes is the Lp space with . An function is equivalently a Lebesgue integrable function once a representative is fixed.
A function on a measure space is a measurable function or such that
Here denotes the essential supremum.
If and are equal almost everywhere, then , so membership in depends only on the equivalence class modulo a null set. These equivalence classes form the case of an space.