integrals of these functions. The lemma is named after Pierre Fatou. Fatou's lemma can be used to prove the Fatou–Lebesgue theorem and Lebesgue's dominated...
24 KB (4,196 words) - 07:51, 29 June 2024
Monotone convergence theorem (redirect from Beppo Levi's lemma)
d\mu .} The proof can also be based on Fatou's lemma instead of a direct proof as above, because Fatou's lemma can be proved independent of the monotone...
24 KB (5,518 words) - 11:56, 1 September 2024
Farkas' lemma Fatou's lemma Gauss's lemma (any of several named after Carl Friedrich Gauss) Greendlinger's lemma Itô's lemma Jordan's lemma Lovász local...
4 KB (402 words) - 00:58, 1 September 2024
The lemma can be viewed as an improvement, in certain settings, of Fatou's lemma to an equality. As such, it has been useful for the study of many variational...
4 KB (689 words) - 19:42, 12 November 2021
0834.01. Fatou conjecture Fatou's theorem Fatou set Fatou–Lebesgue theorem (same as Fatou's lemma) Classification of Fatou components Fatou–Bieberbach...
11 KB (1,236 words) - 09:29, 7 November 2023
[X_{i}].} Fatou's lemma: Let { X n ≥ 0 : n ≥ 0 } {\displaystyle \{X_{n}\geq 0:n\geq 0\}} be a sequence of non-negative random variables. Fatou's lemma states...
52 KB (7,602 words) - 16:21, 31 August 2024
based fundamentally on an application of the triangle inequality and Fatou's lemma. Applied to probability theory, Scheffe's theorem, in the form stated...
3 KB (279 words) - 23:12, 28 April 2024
Wald's lemma Glivenko–Cantelli lemma Neyman–Pearson lemma Robbins lemma Factorization lemma Fatou's lemma Frostman's lemma (geometric measure theory) Malliavin's...
8 KB (524 words) - 11:06, 2 August 2024
theorem is a special case of the Fatou–Lebesgue theorem. Below, however, is a direct proof that uses Fatou’s lemma as the essential tool. Since f is...
13 KB (2,207 words) - 15:47, 23 June 2024
hence integrable. Using linearity of the Lebesgue integral and applying Fatou's lemma to the non-negative functions f n + g {\displaystyle f_{n}+g} we get...
4 KB (645 words) - 16:48, 16 July 2024
if the Riemann integral is replaced by the Lebesgue integral, then Fatou's lemma or the dominated convergence theorem shows that g does satisfy the fundamental...
21 KB (3,357 words) - 14:05, 20 August 2024
)}=1} Almost sure convergence implies convergence in probability (by Fatou's lemma), and hence implies convergence in distribution. It is the notion of...
40 KB (5,268 words) - 10:22, 23 August 2024
differentiation of series of monotonic functions. It can be proven by using Fatou's lemma and the properties of null sets. Assume I ⊆ R {\displaystyle I\subseteq...
1 KB (205 words) - 19:14, 22 December 2022
calculus, the Arzelà-Ascoli theorem, the Stone-Weierstrass theorem, Fatou's lemma, and the monotone convergence and dominated convergence theorems. Various...
49 KB (7,673 words) - 10:05, 6 August 2024
d\mu .} The value of any of the integrals is allowed to be infinite. Fatou's lemma: If {fk}k ∈ N is a sequence of non-negative measurable functions, then...
41 KB (5,848 words) - 18:07, 12 July 2024
then (fn) converges to f locally in measure. The converse is false. Fatou's lemma and the monotone convergence theorem hold if almost everywhere convergence...
7 KB (1,023 words) - 00:40, 30 March 2024
Carathéodory's extension theorem Content (measure theory) Fubini's theorem Fatou's lemma Fuzzy measure theory Geometric measure theory Hausdorff measure Inner...
35 KB (5,554 words) - 13:16, 1 September 2024
{\displaystyle E(X_{n}\mid {\mathcal {H}})\to E(X\mid {\mathcal {H}})} . Fatou's lemma: If E ( inf n X n ∣ H ) > − ∞ {\displaystyle \textstyle E(\inf _{n}X_{n}\mid...
33 KB (5,959 words) - 13:04, 22 April 2024
theorem Cafiero convergence theorem Fatou's lemma Monotone convergence theorem for integrals (Beppo Levi's lemma) Interchange of derivative and integral:...
6 KB (670 words) - 16:01, 1 February 2024
convergence in measure with respect to μ . {\displaystyle \mu .} Then by Fatou's lemma the integral, seen as an operator from L + ( X , μ ) {\displaystyle...
24 KB (3,973 words) - 09:19, 4 September 2024
mathematical results, including Caratheodory-Type Selection Theorems, the Fatou’s Lemma in infinite dimensional spaces, and the upper and lower semicontinuity...
5 KB (449 words) - 17:00, 26 August 2024
Lebesgue's dominated convergence theorem, the Riesz–Fischer theorem, Fatou's lemma, and Fubini's theorem may also readily be proved using this construction...
11 KB (1,647 words) - 14:36, 23 July 2024
continuous if and only if { f } {\displaystyle \{f\}} is equicontinuous. Fatou Fatou's lemma Fourier 1. The Fourier transform of a function f {\displaystyle...
22 KB (3,284 words) - 12:38, 30 August 2024
_{n\to \infty }1_{X_{n}>0}>0} with positive probability. By the reverse Fatou lemma, it suffices to show that inf n → ∞ Pr ( X n > 0 ) > 0 {\displaystyle...
10 KB (1,630 words) - 01:33, 17 July 2023
everywhere, conull set Lp space Borel–Cantelli lemma Lebesgue's monotone convergence theorem Fatou's lemma Absolutely continuous Uniform absolute continuity...
2 KB (221 words) - 02:51, 2 May 2022
as rearrangement inequalities or the Brezis-Lieb lemma which provides the missing term in Fatou's lemma for sequences of functions converging almost everywhere...
30 KB (3,203 words) - 19:55, 20 June 2024
subsequence and, in turn, T f n → T f {\textstyle Tf_{n}\to Tf} a.e. Then, by Fatou’s lemma and recalling that (4) holds true for simple functions, ‖ T f ‖ q θ...
39 KB (6,106 words) - 06:10, 3 May 2023
also contributed to real analysis by developing generalizations of Fatou's lemma and Berge's maximum theorem. Feinberg has also worked on applications...
13 KB (1,203 words) - 23:08, 16 January 2024
results: Lebesgue differentiation theorem Rademacher differentiation theorem Fatou's theorem on nontangential convergence. Fractional integration theorem Here...
10 KB (1,620 words) - 19:40, 19 July 2024
Conditional expectation: law of total expectation, law of total variance Fatou's lemma and the monotone and dominated convergence theorems Markov's inequality...
8 KB (556 words) - 00:09, 23 June 2024