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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 Riesz–Fischer theorem also applies in a more general setting. Let R be an inner product space consisting of functions (for example, measurable functions on the line, analytic functions in the unit disc; in old literature, sometimes called Euclidean Space), and let {} be an orthonormal system in R (e.g. Fourier basis, Hermite or Laguerre polynomials, etc. – see orthogonal polynomials ...
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 ().
Measure and integration (as the English translation of the title reads) is a definitive monograph on integration and measure theory: the treatment of the limiting behavior of the integral of various kind of sequences of measure-related structures (measurable functions, measurable sets, measures and their combinations) is somewhat conclusive.
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.
We take a collection of linear operators {T z : z ∈ S} on the space of simple functions in L 1 (μ 1) into the space of all μ 2-measurable functions on Ω 2. We assume the following further properties on this collection of linear operators:
For (,) a measurable space, a sequence μ n is said to converge setwise to a limit μ if = ()for every set .. Typical arrow notations are and .. For example, as a consequence of the Riemann–Lebesgue lemma, the sequence μ n of measures on the interval [−1, 1] given by μ n (dx) = (1 + sin(nx))dx converges setwise to Lebesgue measure, but it does not converge in total variation.
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 (,).