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In mathematics, the L p spaces are function spaces defined using a natural generalization of the p-norm for finite-dimensional vector spaces.They are sometimes called Lebesgue spaces, named after Henri Lebesgue (Dunford & Schwartz 1958, III.3), although according to the Bourbaki group (Bourbaki 1987) they were first introduced by Frigyes Riesz ().
The classical definition of a locally integrable function involves only measure theoretic and topological [4] concepts and can be carried over abstract to complex-valued functions on a topological measure space (X, Σ, μ): [5] however, since the most common application of such functions is to distribution theory on Euclidean spaces, [2] all ...
Continuous functions, monotone functions, step functions, semicontinuous functions, Riemann-integrable functions, and functions of bounded variation are all Lebesgue measurable. [2] A function f : X → C {\displaystyle f:X\to \mathbb {C} } is measurable if and only if the real and imaginary parts are measurable.
The simplest example of a direct integral are the L 2 spaces associated to a (σ-finite) countably additive measure μ on a measurable space X. Somewhat more generally one can consider a separable Hilbert space H and the space of square-integrable H-valued functions (,).
The function F defined on the unit disk by F(re iθ) = (f ∗ P r)(e iθ) is harmonic, and M f is the radial maximal function of F. When M f belongs to L p (T) and p ≥ 1, the distribution f "is" a function in L p (T), namely the boundary value of F. For p ≥ 1, the real Hardy space H p (T) is a subset of L p (T).
The integral of a non-negative general measurable function is then defined as an appropriate supremum of approximations by simple functions, and the integral of a (not necessarily positive) measurable function is the difference of two integrals of non-negative measurable functions.
Let L 2 (X, μ) be the space of those complex-valued measurable functions on X for which the Lebesgue integral of the square of the absolute value of the function is finite, i.e., for a function f in L 2 (X, μ), | | <, and where functions are identified if and only if they differ only on a set of measure zero.
is a function space.Its elements are the essentially bounded measurable functions. [2]More precisely, is defined based on an underlying measure space, (,,). Start with the set of all measurable functions from to which are essentially bounded, that is, bounded except on a set of measure zero.