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lim inf X n consists of elements of X which belong to X n for all except finitely many n (i.e., for cofinitely many n). That is, x ∈ lim inf X n if and only if there exists some m > 0 such that x ∈ X n for all n > m. Observe that x ∈ lim sup X n if and only if x ∉ lim inf X n c.
Let f 1, f 2, ... denote a sequence of real-valued measurable functions defined on a measure space (S,Σ,μ).If there exists a Lebesgue-integrable function g on S which dominates the sequence in absolute value, meaning that |f n | ≤ g for all natural numbers n, then all f n as well as the limit inferior and the limit superior of the f n are integrable and
This sequence converges uniformly on S to the zero function and the limit, 0, is reached in a finite number of steps: for every x ≥ 0, if n > x, then f n (x) = 0. However, every function f n has integral −1. Contrary to Fatou's lemma, this value is strictly less than the integral of the limit (0).
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In mathematics, the limit of a sequence of sets,, … (subsets of a common set ) is a set whose elements are determined by the sequence in either of two equivalent ways: (1) by upper and lower bounds on the sequence that converge monotonically to the same set (analogous to convergence of real-valued sequences) and (2) by convergence of a sequence of indicator functions which are themselves ...
() (using x ≥ 0 to obtain the final inequality) so that: = One must use lim sup because it is not known if t n converges. For the other inequality, by the above expression for t n , if 2 ≤ m ≤ n , we have: 1 + x + x 2 2 !
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The set lim sup E n is sometimes denoted {E n i.o.}, where "i.o." stands for "infinitely often". The theorem therefore asserts that if the sum of the probabilities of the events E n is finite, then the set of all outcomes that are "repeated" infinitely many times must occur with probability zero.