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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.
In mathematics, strong convergence may refer to: The strong convergence of random variables of a probability distribution. The norm-convergence of a sequence in a Hilbert space (as opposed to weak convergence). The convergence of operators in the strong operator topology
This is the “weak convergence of laws without laws being defined” — except asymptotically. [1] In this case the term weak convergence is preferable (see weak convergence of measures), and we say that a sequence of random elements {X n} converges weakly to X (denoted as X n ⇒ X) if
The definition of weak convergence can be extended to Banach spaces. A sequence of points ( x n ) {\displaystyle (x_{n})} in a Banach space B is said to converge weakly to a point x in B if f ( x n ) → f ( x ) {\displaystyle f(x_{n})\to f(x)} for any bounded linear functional f {\displaystyle f} defined on B {\displaystyle B} , that is, for ...
Convergence in the Radon metric implies weak convergence of measures: (,), but the converse implication is false in general. Convergence of measures in the Radon metric is sometimes known as strong convergence, as contrasted with weak convergence.
In mathematics, weak convergence may refer to: Weak convergence of random variables of a probability distribution; Weak convergence of measures, of a sequence of probability measures; Weak convergence (Hilbert space) of a sequence in a Hilbert space more generally, convergence in weak topology in a Banach space or a topological vector space
In the mathematical fields of differential geometry and geometric analysis, inverse mean curvature flow (IMCF) is a geometric flow of submanifolds of a Riemannian or pseudo-Riemannian manifold. It has been used to prove a certain case of the Riemannian Penrose inequality , which is of interest in general relativity .
In analysis, a topology is called strong if it has many open sets and weak if it has few open sets, so that the corresponding modes of convergence are, respectively, strong and weak. (In topology proper, these terms can suggest the opposite meaning, so strong and weak are replaced with, respectively, fine and coarse.)