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Lebesgue spaces

Lebesgue spaces are function spaces associated to measure spaces (X, M, μ), where X is a set, M is a σ-algebra of subsets of X, and μ is a countably additive measure on M. Let L (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 (X, μ), ∫ X | f | 2 d μ < ∞, {\displaystyle \int _{X}|f|^{2}\mathrm {d} \mu <\infty \,,} and where functions are identified if and only if they differ only on a set of measure zero.

The inner product of functions f and g in L (X, μ) is then defined as ⟨ f, g ⟩ = ∫ X f (t) g (t) ¯ d μ (t) {\displaystyle \langle f,g\rangle =\int _{X}f(t){\overline {g(t)}}\,\mathrm {d} \mu (t)} or ⟨ f, g ⟩ = ∫ X f (t) ¯ g (t) d μ (t), {\displaystyle \langle f,g\rangle =\int _{X}{\overline {f(t)}}g(t)\,\mathrm {d} \mu (t)\,,}

where the second form (conjugation of the first element) is commonly found in the theoretical physics literature. For f and g in L, the integral exists because of the Cauchy–Schwarz inequality, and defines an inner product on the space. Equipped with this inner product, L is in fact complete. The Lebesgue integral is essential to ensure completeness: on domains of real numbers, for instance, not enough functions are Riemann integrable.

The Lebesgue spaces appear in many natural settings. The spaces L (R) and L () of square-integrable functions with respect to the Lebesgue measure on the real line and unit interval, respectively, are natural domains on which to define the Fourier transform and Fourier series. In other situations, the measure may be something other than the ordinary Lebesgue measure on the real line. For instance, if w is any positive measurable function, the space of all measurable functions f on the interval satisfying ∫ 0 1 | f (t) | 2 w (t) d t < ∞ {\displaystyle \int _{0}^{1}{\bigl |}f(t){\bigr |}^{2}w(t)\,\mathrm {d} t<\infty } is called the weighted L space L (), and w is called the weight function. The inner product is defined by ⟨ f, g ⟩ = ∫ 0 1 f (t) g (t) ¯ w (t) d t. {\displaystyle \langle f,g\rangle =\int _{0}^{1}f(t){\overline {g(t)}}w(t)\,\mathrm {d} t\,.}

The weighted space L () is identical with the Hilbert space L (, μ) where the measure μ of a Lebesgue-measurable set A is defined by μ (A) = ∫ A w (t) d t. {\displaystyle \mu (A)=\int _{A}w(t)\,\mathrm {d} t\,.}

Weighted L spaces like this are frequently used to study orthogonal polynomials, because different families of orthogonal polynomials are orthogonal with respect to different weighting functions.