• 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
  • 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 (5,489 words) - 16:58, 27 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
  • 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...
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  • based fundamentally on an application of the triangle inequality and Fatou's lemma. Applied to probability theory, Scheffe's theorem, in the form stated...
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  • Thumbnail for Pierre Fatou
    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) - 05:10, 27 September 2024
  • Thumbnail for Expected value
    [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 (7,614 words) - 02:30, 29 September 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 (2,208 words) - 07:57, 17 October 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...
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  • 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...
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  • 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...
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  • Thumbnail for Antiderivative
    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...
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  • Wald's lemma Glivenko–Cantelli lemma Neyman–Pearson lemma Robbins lemma Factorization lemma Fatou's lemma Frostman's lemma (geometric measure theory) Malliavin's...
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  • theorem Cafiero convergence theorem Fatou's lemma Monotone convergence theorem for integrals (Beppo Levi's lemma) Interchange of derivative and integral:...
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  • )}=1} Almost sure convergence implies convergence in probability (by Fatou's lemma), and hence implies convergence in distribution. It is the notion of...
    41 KB (5,280 words) - 21:51, 25 September 2024
  • Thumbnail for Lebesgue integral
    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 (5,861 words) - 06:24, 5 October 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...
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  • 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) - 19:32, 28 October 2024
  • Thumbnail for Measure (mathematics)
    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) - 21:47, 26 October 2024
  • everywhere, conull set Lp space Borel–Cantelli lemma Lebesgue's monotone convergence theorem Fatou's lemma Absolutely continuous Uniform absolute continuity...
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  • Thumbnail for Semi-continuity
    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 (3,980 words) - 23:40, 18 September 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
  • 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...
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  • Thumbnail for Elliott H. Lieb
    as rearrangement inequalities or the Brezis-Lieb lemma which provides the missing term in Fatou's lemma for sequences of functions converging almost everywhere...
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  • continuous selection theorems, Caratheodory-Type Selection Theorems, the Fatou’s Lemma in infinite dimensional spaces, fixed points for discontinuous correspondences...
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  • Conditional expectation: law of total expectation, law of total variance Fatou's lemma and the monotone and dominated convergence theorems Markov's inequality...
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  • Thumbnail for Eugene A. Feinberg
    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...
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  • 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) - 13:36, 29 October 2024
  • Thumbnail for Czesław Olech
    ISBN 978-3-540-53120-3. Olech, Czeslaw (1987). "Onn-dimensional extensions of Fatou's lemma". Zeitschrift für Angewandte Mathematik und Physik. 38 (2): 266–272...
    15 KB (1,542 words) - 13:14, 19 October 2024