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Short-term memory has limited capacity and is often referred to as "working-memory", however these are not the same. Working memory involves a different part of the brain and allows you to manipulate it after initial storage. The information that travels from sensory memory to short-term memory must pass through the Attention gateway. The ...
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.
is a function space.Its elements are the essentially bounded measurable functions. [2]More precisely, is defined based on an underlying measure space, (,,). Start with the set of all measurable functions from to which are essentially bounded, that is, bounded except on a set of measure zero.
For this reason, the Lebesgue definition makes it possible to calculate integrals for a broader class of functions. For example, the Dirichlet function, which is 1 where its argument is rational and 0 otherwise, has a Lebesgue integral, but does not have a Riemann integral. Furthermore, the Lebesgue integral of this function is zero, which ...
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 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 ...
The above example motivates the general definition: A family of bounded operators {T x} x∈ X with T x ∈ L(H x) is said to be strongly measurable if and only if its restriction to each X n is strongly measurable. This makes sense because H x is constant on X n. Measurable families of operators with an essentially bounded norm, that is
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 ().