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The problem is a differential equation of the form [()] + = for an unknown function y on an interval [a, b], satisfying general homogeneous Robin boundary conditions {() + ′ ′ = + ′ ′ =. The functions p, q, and w are given in advance, and the problem is to find the function y and constants λ for which the equation has a solution.
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.
The vector space of (equivalence classes of) measurable functions on (,,) is denoted (,,) (Kalton, Peck & Roberts 1984). By definition, it contains all the , and is equipped with the topology of convergence in measure.
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 ...
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 ...
If A is a Lebesgue-measurable set with λ(A) = 0 (a null set), then every subset of A is also a null set. A fortiori, every subset of A is measurable. If A is Lebesgue-measurable and x is an element of R n, then the translation of A by x, defined by A + x = {a + x : a ∈ A}, is also Lebesgue-measurable and has the same measure as A.
Let (,,) be a measure space, i.e. : [,] is a set function such that () = and is countably-additive. All functions considered in the sequel will be functions :, where = or .We adopt the following definitions according to Bogachev's terminology.
The measure class [clarification needed] of μ and the measure equivalence class of the multiplicity function x → dim H x completely characterize the projection-valued measure up to unitary equivalence. A projection-valued measure π is homogeneous of multiplicity n if and only if the multiplicity function has constant value n. Clearly, Theorem.