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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.
In mathematical analysis, a Carathéodory function (or Carathéodory integrand) is a multivariable function that allows us to solve the following problem effectively: A composition of two Lebesgue-measurable functions does not have to be Lebesgue-measurable as well. Nevertheless, a composition of a measurable function with a continuous function ...
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 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.
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 left adjoint: of the adjunction above is the identity on objects, and on morphisms it gives the canonical Markov kernel induced by a measurable function described above. As mentioned above, one can construct a category of probability spaces and measure-preserving Markov kernels as the slice category ( H o m S t o c h ( 1 , − ) , S t o c h ...
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.
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.