• field of real analysis, the monotone convergence theorem is any of a number of related theorems proving the good convergence behaviour of monotonic sequences...
    24 KB (5,328 words) - 13:51, 31 May 2025
  • martingale convergence theorem is a random variable analogue of the monotone convergence theorem, which states that any bounded monotone sequence converges. There...
    17 KB (2,800 words) - 06:33, 14 April 2025
  • on N {\displaystyle N} . Fatou's lemma does not require the monotone convergence theorem, but the latter can be used to provide a quick and natural proof...
    28 KB (5,120 words) - 05:53, 25 April 2025
  • (following from Brouwer's bar theorem) and is strong enough to give short proofs of key theorems. The monotone convergence theorem (described as the fundamental...
    11 KB (1,511 words) - 11:05, 6 January 2025
  • there exists a monotone subsequence, likewise also bounded. It follows from the monotone convergence theorem that this subsequence converges. The general...
    12 KB (2,066 words) - 16:16, 24 May 2025
  • Thumbnail for Lebesgue integral
    take limits under the integral sign (via the monotone convergence theorem and dominated convergence theorem). While the Riemann integral considers the area...
    41 KB (5,918 words) - 20:43, 16 May 2025
  • sufficient condition for the convergence of expected values of random variables. Lebesgue's dominated convergence theorem. Let ( f n ) {\displaystyle (f_{n})}...
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  • Thumbnail for Expected value
    convergence results specify exact conditions which allow one to interchange limits and expectations, as specified below. Monotone convergence theorem:...
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  • rules and bargaining systems. Monotone class theorem, in measure theory Monotone convergence theorem, in mathematics Monotone polygon, a property of a geometric...
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  • Thumbnail for Integral test for convergence
    mathematics, the integral test for convergence is a method used to test infinite series of monotonic terms for convergence. It was developed by Colin Maclaurin...
    10 KB (1,727 words) - 01:02, 15 November 2024
  • f dμ ≤ dν. The supremum of all such functions, along with the monotone convergence theorem, then furnishes the Radon–Nikodym derivative. The fact that the...
    23 KB (3,614 words) - 20:46, 30 April 2025
  • mathematical field of analysis, Dini's theorem says that if a monotone sequence of continuous functions converges pointwise on a compact space and if the...
    5 KB (716 words) - 12:47, 28 March 2024
  • Thumbnail for Monotonic function
    In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept...
    19 KB (2,471 words) - 01:32, 25 January 2025
  • it follows from the monotone convergence theorem for series that the sum of this infinite series is equal to e. The binomial theorem is closely related...
    42 KB (6,735 words) - 17:25, 22 May 2025
  • {\displaystyle p<\infty ,} the Minkowski inequality and the monotone convergence theorem imply that ∫ ( ∑ n = 0 ∞ | u n | ) p d μ ≤ ( ∑ n = 0 ∞ ‖ u n...
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  • functions on ( X , Σ ) {\displaystyle (X,\Sigma )} , by Lebesgue's monotone convergence theorem ν {\displaystyle \nu } can be shown to correspond to an L 1 (...
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  • integral Lebesgue integration Monotone convergence theorem – relates monotonicity with convergence Intermediate value theorem – states that for each value...
    14 KB (1,603 words) - 13:55, 14 September 2024
  • Alternating series test (category Convergence tests)
    monotonically decreasing sequence S2m+1, the monotone convergence theorem then implies that this sequence converges as m approaches infinity. Similarly, the...
    12 KB (1,828 words) - 20:10, 23 May 2025
  • follows using the monotone convergence theorem. By assumption (3), the infinite sequence on the right-hand side of (15) converges, hence SN is integrable...
    25 KB (3,188 words) - 01:42, 27 April 2024
  • _{s=0}^{\infty }|X_{s+1}-X_{s}|\cdot \mathbf {1} _{\{\tau >s\}}} . By the monotone convergence theorem E [ M ] = E [ | X 0 | ] + ∑ s = 0 ∞ E [ | X s + 1 − X s | ⋅ 1...
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  • Dirichlet function provides a counterexample showing that the monotone convergence theorem is not true in the context of the Riemann integral. Proof Using...
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  • sequences and modes of convergence, and different proof techniques may be more appropriate than others for proving each type of convergence of each type of sequence...
    11 KB (1,758 words) - 14:24, 4 September 2024
  • theorem, the Stone-Weierstrass theorem, Fatou's lemma, and the monotone convergence and dominated convergence theorems. Various ideas from real analysis...
    49 KB (7,668 words) - 20:26, 29 May 2025
  • Vitali convergence theorem Fichera convergence theorem Cafiero convergence theorem Fatou's lemma Monotone convergence theorem for integrals (Beppo Levi's lemma)...
    6 KB (670 words) - 17:46, 20 November 2024
  • In mathematics, the Stolz–Cesàro theorem is a criterion for proving the convergence of a sequence. It is named after mathematicians Otto Stolz and Ernesto...
    17 KB (4,268 words) - 13:48, 17 March 2025
  • contained in the bounded set C 0 {\displaystyle C_{0}} . The monotone convergence theorem for bounded sequences of real numbers now guarantees the existence...
    8 KB (1,565 words) - 17:42, 13 September 2024
  • preparation theorem (singularity theory) Mean value theorem (calculus) Monotone convergence theorem (mathematical analysis) Müntz–Szász theorem (functional...
    78 KB (6,293 words) - 12:16, 2 May 2025
  • well-known theorem describes the set of discontinuities of a monotone real-valued function of a real variable; all discontinuities of such a (monotone) function...
    28 KB (3,495 words) - 19:16, 14 May 2025
  • {a_{n}}}}}}}\right)} is monotonically increasing. Therefore it converges, by the monotone convergence theorem. If the sequence ( a 1 + a 2 + ⋯ a n ) {\displaystyle...
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  • course at the United States Marine Corps School of Infantry Monotone convergence theorem, in mathematics Motivation crowding theory, in economics Member...
    3 KB (379 words) - 15:52, 3 September 2023