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  2. Measurable function - Wikipedia

    en.wikipedia.org/wiki/Measurable_function

    A Lebesgue measurable function is a measurable function : (,) (,), where is the -algebra of Lebesgue measurable sets, and is the Borel algebra on the complex numbers. Lebesgue measurable functions are of interest in mathematical analysis because they can be integrated.

  3. Convergence of measures - Wikipedia

    en.wikipedia.org/wiki/Convergence_of_measures

    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.

  4. Lp space - Wikipedia

    en.wikipedia.org/wiki/Lp_space

    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 ().

  5. Convergence in measure - Wikipedia

    en.wikipedia.org/wiki/Convergence_in_measure

    If X = [a,b] ⊆ R and μ is Lebesgue measure, there are sequences (g n) of step functions and (h n) of continuous functions converging globally in measure to f. If f and f n (n ∈ N) are in L p (μ) for some p > 0 and (f n) converges to f in the p-norm, then (f n) converges to f globally in measure. The converse is false.

  6. Riesz–Fischer theorem - Wikipedia

    en.wikipedia.org/wiki/Riesz–Fischer_theorem

    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 ...

  7. Locally integrable function - Wikipedia

    en.wikipedia.org/wiki/Locally_integrable_function

    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.

  8. Vitali convergence theorem - Wikipedia

    en.wikipedia.org/wiki/Vitali_convergence_theorem

    When () <, a set of functions (,,) is uniformly integrable if and only if it is bounded in (,,) and has uniformly absolutely continuous integrals. If, in addition, μ {\displaystyle \mu } is atomless, then the uniform integrability is equivalent to the uniform absolute continuity of integrals.

  9. Hölder's inequality - Wikipedia

    en.wikipedia.org/wiki/Hölder's_inequality

    As above, let f and g denote measurable real- or complex-valued functions defined on S. If ‖ fg ‖ 1 is finite, then the pointwise products of f with g and its complex conjugate function are μ-integrable, the estimate