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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. 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 ...

  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. 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.

  6. Square-integrable function - Wikipedia

    en.wikipedia.org/wiki/Square-integrable_function

    In mathematics, a square-integrable function, also called a quadratically integrable function or function or square-summable function, [1] is a real- or complex-valued measurable function for which the integral of the square of the absolute value is finite.

  7. Conditional expectation - Wikipedia

    en.wikipedia.org/wiki/Conditional_expectation

    for -measurable , we have ((())) =, i.e. the conditional expectation () is in the sense of the L 2 (P) scalar product the orthogonal projection from to the linear subspace of -measurable functions. (This allows to define and prove the existence of the conditional expectation based on the Hilbert projection theorem .)

  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. Riesz–Markov–Kakutani representation theorem - Wikipedia

    en.wikipedia.org/wiki/Riesz–Markov–Kakutani...

    where α(x) is a function of bounded variation on the interval [0, 1], and the integral is a Riemann–Stieltjes integral. Since there is a one-to-one correspondence between Borel regular measures in the interval and functions of bounded variation (that assigns to each function of bounded variation the corresponding Lebesgue–Stieltjes measure ...