dominated convergence theorem example

(b) Converge in Lp)converge in Lq)converge in probability ) converge weakly, p q 1. Hence,forcharacters˜p 1q p 1q p 1qk,wehavetheFourierexpansion # kp! Prove the Theorem.-- State Fatou's Lemma. Dominated Convergence Theorem Involving Small Riemann Sums on E. Suppose that there exists a Lebesgue integrable function g on E such that |f n| ≤ g for all n. Then f is Lebesgue integrable and lim Z E |f n −f| = 0. Since we hypothesize limn→∞ R E gn = R E g < ∞, then for index n suﬃciently large, R E gn < ∞. Its power and utility are two of the primary theoretical advantages of Lebesgue integration over Riemann A DOMINATED CONVERGENCE THEOREM FOR EISENSTEIN SERIES 7 Proof. Remark For the direct proof of this theorem, you can see Theorem 3.9.1 on Durrett's book, or the section on weak convergence of Billingsley's book. So when does a.s. convergence imply convergence in distribution: need to control for the cases where things go really wrong with small probability. It is more convenient to apply the following dominated convergence theorem for inﬁnite sums. Examples of Lebesgue dominated convergence theorem First, suppose that 1 p<1. In this paper, we deﬁne two interval functions U . A short summary of this paper. If limn→∞ fn = f a.e. A.s. convergence implies convergence in probability. ﬁrst example and g(x) = 0 in the second example fails to capture the graphs of any g n(x). Sometimes these sequences of functions con-verge or get very close to another function. Then y(x) in Theorem 3.30 can be re- placed by a positive constant M. As an application of the dominated convergence theorem (Theorem 3.30), we can show the validity of interchanging a derivative with an integral less restrictive than elementary calculus courses. The Pythagorean Theorem is a starting place for trigonometry, which leads to methods, for example, for calculating length of a lake. Bruce K. Driver Analysis Tools with Examples June 30, 2004 File:anal.tex Springer Berlin Heidelberg NewYork HongKong London Milan Paris Tokyo 4 2017-18 Mathematics MA2224 Proposition 3.2.4. Lebesgue's dominated convergence theorem is a special case of the Fatou-Lebesgue theorem. Claudio Sacerdoti Coen. Lebesgue integral. ground information in measure theory and convergence is provided and several examples are considered. Answer: Almost sure convergence should imply convergence of integrals unless the functions you're trying to integrate are behaving erratically. Let (f n) be a sequence of complex-valued measurable functions on a measure space (S, Σ, μ).Suppose that the sequence converges pointwise to a function f and is dominated by some integrable function g in the sense that [math]\displaystyle{ |f_n(x)| \le g(x) }[/math] for all numbers n in the index set of the sequence and all points x ∈ S. Below, however, is a direct proof that uses Fatou's lemma as the essential tool. Statement. 1.7 Theorem. Lebesgue's dominated convergence theorem is a special case of the Fatou-Lebesgue theorem. ∈ M+, if fn(x) → f(x) for n → ∞ for all x ∈ X then if there is a g ∈ M+ with R gdµ < ∞ such that fn ≤ g then R fndµ → R fdµ for n → ∞.. - p.1/21 Example Consider the measure space (R,B,m) and the function fn(x) = n √ x 1 +n2x2 x > 0. However, we ﬁrst restate the dominated convergence theorem, ﬁrst discussed in Problem 11.1, since we will often use this result in verifying the Lindeberg condition. Relationships between convergence: (a) Converge a.c. )converge in probability )weak convergence. About Press Copyright Contact us Creators Advertise Developers Terms Privacy Policy & Safety How YouTube works Test new features Press Copyright Contact us Creators . (Or by Corollary 6.12 on page 71.) Lebesgue's dominated convergence theorem is a special case of the Fatou-Lebesgue theorem. Xn d! Theorem 11.2 If F n(x) and F(x) are cdf's and F(x) is continuous, then pointwise . Example 11.4 (Proof of Theorem 11.2). X ) g . Since f is the pointwise limit of the sequence ( f n ) of measurable functions that are dominated by g , it is also measurable and dominated by g , hence it is integrable. We know that dominated convergence and monotone conver gence still hold if we replace convergence with convergence almost ev erywhere. Assume f: R R !R is such that x7!f[t](x) = : Thus E(X)t+ E(X 2) t 2 2! Stokes' and Gauss' Theorems - Penn Math . If you can uniformly bound all the functions in your sequence by an integrable function, then your sequence of functions can behave no more erratically . For example, if M is non-measurable in Y and if N is a null set in X, the N × M is a null set in X × Y . Corollary 4. If ff Thus, it is clear that pointwise convergence does not in general imply uniform convergence. on [a;b]. We shall use again Theorem A.5.1. Arzela-Lebesgue dominated convergence theorem follows then rather easily. Among such non-trivial examples there are Lebesgue filters (for which the Lebesgue dominated convergence theorem is valid), Egorov filters which correspond to the Egorov theorem on almost uniform convergence [7], and those filters F for which every weakly F convergent sequence has a norm-bounded subsequence [6]. Xn d! Show that lim n!1 logn Xn k=1 1 k = lim n!1 Z n 0 1 x x n n logxdx= Z 1 0 e logxdx: Solution. To this aim, let us recall that there exist mD > 0 and m ℱ 0 such that. 1. Let ff ngbe a sequence of integrable functions converging to f a.e. Then fn(x) → 0 for n → ∞ for . F 1 N1 ˜ b! Examples of Lebesgue dominated convergence theorem. The martingale convergence theorems, first formulated by Joseph Doob, are among the most important results in the theory of martingales. A number of elementary proofs were published by Enrico Zoli. If fn ≥ 0, then R liminfn fndν ≤ liminfn R fndν. Show activity on this post. Share Monotone Convergence Theorem(MON): If X n a:s:!X and X n is increasing almost surely, then . The above theorem can be combined with things like monotone convergence theorem or Lebesgue dominated convergence theorem to switch the limits and conclude that E(X ˝) = E(X 0). (c) Convergence in KL divergence )Convergence in total variation)strong convergence of measure )weak convergence, where i. n If fn is a sequence such that jfnj g, and fn!f pointwise, then Z f = lim n!1 Z fn! Other results in Section 2 give necessary and sufficient conditions for ?in\fn- f\v -> 0 and for finfn ?>/if. The Continuous Mapping Theorem Let g be continuous on a set C where P(X 2 C) = 1. Dominated Convergence Theorem (using both the Monotone Convergence Theorem and the Bounded Convergence Theorem). Let us first determine the pointwise limit of the sequence of functions: Therefore converges to when n goes to infinity.. Let us show now that is integrable on . 4.32. This question shows research effort; it is useful and clear. Lebesgue Dominated Convergence Theorem Suppose g 0 is non-negative, measurable, and R g <1. But avoid …. A measurable function f deﬁned on E has Locally Small Riemann Sums, or the LSRS property, if for every > 0 there is a positive function δ such that Remark 4.3.2. This state of affairs may account for the fact that the search for an "elementary proof", roughly meaning, independent of the theory of Lebesgue measure, for Arzel'a's theorem is still on. Dominated Convergence Theorem, for any t2R, X1 n=0 tn n! example, [1, 2, 4, 6]. Theorem 1.5 (The Dominated Convergence Theorem). The dominated convergence theorem: If f1,f2,. a quantitative version of the dominated convergence theorem, a mild strengthening of Theorem A.2 of Tao [26], again with an explicit description of the computation of one metastable bound from the other. Theorem 5. Example 3. Without loss of . Let f = fE R : f 1(E) 2Lg: We claim that f is a ˙-algebra. 95. i. We compare Riemann and Lebesgue integration and develop several im-portant theorems. Read Paper. Let ( f n) be a sequence of real-valued measurable functions on a measure space ( X, Σ, μ). and there exists an integrable function g such that |fn| ≤ g a.e., then R limn→∞ fndν = limn→∞ R fndν. Convergence in mean square: Xk -+ X in m.s. If Xand Y are integrable random variables, then (iii) (Monotone . Proof. This theorem is from calculus (or mathematical analysis) and is not easy to apply because we need to show the uniform convergence of an inﬁnite series. Applications can be found in any book on the Lebesgue integral. Below, however, is a direct proof that uses Fatou's lemma as the essential tool. We leave the proof to the reader. Next we look at three natural examples of subgaussian random variables. (ii) (Dominated convergence theorem). So to use Dominated Convergence you now need to figure out an f ( x). This paper. Consider the sequence of functions defined on by. Non-archimedean function spaces and the Lebesgue dominated convergence theorem∗ J. Ka¸kol C. Perez-Garcia W. Sliwa´ Abstract Let M(X,K)be the non-archimedean Banach space of all additive and bounded K-valued measures on the ring of all clopen subsets of a zero- dimensional compact space X, where K is a non-archimedean non-trivially valued complete ﬁeld. You can also prove it by using Skorokhod's Representation Theorem given below: Theorem 18.2 (Skorokhod's Representation Theorem). This question does not show any research effort; it is unclear or not useful. First, let us observe that, by virtue of Lebesgue dominated convergence theorem, it suffices to show that Q(D, ℱ) is relatively compact in L1 ( a, b; X) and bounded in L∞ ( a, b; X ). Key words : Convergence of integrals, dominated convergence theorem, point processes, random measures, mixtures and images of measures, thinning of point processes. Bookmark this question. Counterexamples around Lebesgue's Dominated Convergence Theorem September 11, 2016 Jean-Pierre Merx Leave a comment Let's recall Lebesgue's Dominated Convergence Theorem. What is the role of the Monotone Class Lemma in the proof of Lemma 10.8? 2. In other words, if one has a sequence (f n)∞ =1 of integrable functions, and if f is some . From English to Math Convergence theorems: monotone, dominated Easy, natural examples show that pointwise limits f = lim nf n of measurable functions f n, while still measurable, need not satisfy R f= lim R f n. That is, this failure is not a pathology, but, rather, is completely reasonable. Un-less we say otherwise, we use the narrow topology on P(Rd): the coarsest topology such that for each f2C b(Rd . Suppose that the sequence converges pointwise to a function f and is dominated by some integrable function g in the sense that jfn(x)j g(x); for all numbers n in the index set and all points x 2S. Suppose X n!D X, all taking values in metric space Lp completeness follows easily. Theorem 1.4 Let {f n} be a sequence of measurable functions on E that converges to f a.e. What does (10.5) mean if some of the integrals in (10.4) are $+\infty$? The follow corollary of the dominated convergence theorem gives a condition for the interchange of infinite sum and integral. 5. Dominated convergence theorem holds for convergence in mea sure. The third convergence theorem is the Lebesgue dominated convergence theorem. Give an example that the inequalty in the Lemma might be strict.-- State the Monotone Convergence Theoerem. Annals of Pure and Applied Logic, 2012. Corollary 3.31. E(Xn) = EetX eb2t2=2 = X1 n=0 b2nt2n 2nn! Please be sure to answer the question.Provide details and share your research! almost everywhere, so by the dominated convergence theorem lim n!1 Z X jf f njp d = Z X lim n!1 jf f njp d = 0: Thus f n!fin Lp. (i) R lim n!1f n= lim n!1 R f n is an equivalent statement. F N2;˝q 2N 1 k 2 ‚8 m 1 d| dk 1pF N 2 d 2 N qpdq F N 1 F 1 N 1 d 1 N ˜ m d qm{N 2N1 k 2 ‚8 m 1 d| dk 1pF . Sup-pose that fXng n2N is a sequence of random variables in Lp, where p 1, which converges to X 2L0 in probability. ) must grow faster than a polynomial at ∞and the condition R p∗(x . the dominated convergence theorem implies that Z f Xn k=1 g k p d !0 as n!1; meaning that P 1 k=1 g k converges to fin L p. The following theorem implies that Lp(X) equipped with the Lp-norm is a Banach space. 37 Full PDFs related to this paper. Convergence almost surely: xk -+ X a.s. (or Xk -+ X with probability 1) Convergence in probability: xk -+ X in pr. 1. Download PDF. X = X(o) in the convergence below is often a constant in practice (e.g., with the constant being 0, a statement that xk -+ X in some probabilistic sense reduces to xk -+ 0 in the same sense). Convergence in rth mean also implies convergence in probability. For (2), the relation 0 g n(x) is clear, since 1 cos(nx) 1. 4 1.2 General random variables Key properties of expectation for general random variables are contained in our next theorem. Problem 3. Fatou's Lemma, the Monotone Convergence Theorem and the Dominated Convergence Theorem Consider integration on $\mathbb{R}^d$ as a mapping from the vector space of measurable functions on $\mathbb{R}^d$ to $[-\infty,\infty]$. In particular, we . version of the dominated convergence theorem: Proposition 12.9 (Improved dominated-convergence theorem). Here are some examples: Example 2.7. 3. By a theorem of Euler we have f n(x) !e x for each x, so since Z 1 0 e xlog dx<1 Theorem 14.3 Dominated convergence theorem . Then the Lebesgue Dominated Convergence theorem implies that Z X . Let f(x) be a bounded measurable function defined over a (Lebesgue) measurable set E of finite measure (for intuitive insight view f(x) as the function y = f(x) shown in Fig. Proof. Since f is the pointwise limit of the sequence ( f n ) of measurable functions that are dominated by g , it is also measurable and dominated by g , hence it is integrable. Convergence in probability implies convergence in law. Then ff ngconverges in Lp to some measurable . Below, however, is a direct proof that uses Fatou's lemma as the essential tool. Use the Theorem to show that the Riemann improper integral of 1/x^2 in [1,infinity) is equal to the Lebesgue integral of 1/x^2 in [1,infinity . The example following Fatou's lemma also shows that the assumption about the existence of the dominating function gcan't be dispensed with. 2. (Actually, the Monotone and Dominated Convergence Theorems are also true, in appropriate forms, for Riemann integrals. Let f Note that . The notion of a -uniform bound on the "-metastable pointwise convergence of a View Lecture_25.pdf from MATH 3033 at The Hong Kong University of Science and Technology. Then, the following statements are equivalent: 1.the sequence fjXj n pg n2N is uniformly integrable, 2.Xn Lp!X, and 3. jjXnjj . The Monotone Convergence Theorem and Dominated Convergence Theorem could be stated without proof, and then examples and exercises can be given. This convergence theorem does not require monotonicity of the sequence (f k)1 k=1 of in-tegrable functions, but only that there is an L1 function gthat dominates the pointwise a.e.convergent sequence (f k)1 k=1, i.e., jf kj gfor all k. First we'll look at a counterexample to see why "domination" is a necessary condition, and we'll close by using the DCT to compute lim n→∞∫R nsin(x/n) x(x2+1). (i) (Fatou's lemma). By Theorem 1.1, E(X n+ Y n) = EX n+ EY n: Letting n!1, by the virtue of Lebesgue's monotone convergence theorem, we get in the limit E(X+ Y) = EX+ EY. The first martingale convergence theorem states that if the expected absolute value is bounded in the time, then the martingale process converges with probability 1. Theorem 1.1. Thanks for contributing an answer to Mathematics Stack Exchange! Let u(x) < 0. by the dominated convergence theorem, with dominating functions g(x) = M1 0 x a. Theorem 6. Theorem 0.4. be a sequence of Borel functions on (Ω,F,ν). Arzela S Dominated Convergence Theorem For The Riemann-PDF Free Download. The Fatou-Lebesgue Theorem deals with inequalities, involving lim inf and lim sup, rather than equalities as was the case in the former two theorems. If f: R !R is Lebesgue measurable, then f 1(B) 2L for each Borel set B. Roughly speaking, a "convergence theorem" states that integrability is preserved under taking limits. Lebesgue's dominated convergence theorem. (Take this sequence for example.) -- State the Lebesgue Dominated Convergence Theorem. ∫ g ( x) d μ = ∫ [ 0, 1] 1 x d μ = 2 x | 0 1 = 2 < ∞ by Lebesgue's Dominated Convergence Theorem you know that if f ( x) equals the pointwise limit of the f n ( x) almost everywhere on [ 0, 1], then lim n → ∞ ∫ [ 0, 1] n sin ( x) 1 + n 2 x d μ = ∫ [ 0, 1] f ( x) d μ. In this paper we prove a dominated convergence theorem for Eisenstein series. Proof. The most natural example of a subgaussian random variable is that of a centered Gaussian. Remark. If fX ngis a martingale and ˝ a stopping time so that P(˝ < 1) = 1 and E(jX ˝j) <1;and lim n!1E(X n1 ˝>n) = 0, then E . Then, 1. MATH3033 — Real Analysis Lecture 25 — 30th November 2021 Review: (1) Monotone Convergence Theorem Lebesgue's Dominated Convergence Theorem Let ffngbe a sequence of real-valued measurable functions on a measure space (S; ; ). Bounded, dominated, monotone convergence theorems. 8.5.2 Dominated Convergence We will use Fatou's Lemma to obtain the dominated convergence theorem of Lebesgue. However, the following theorem gives a special case in which it does. Combining these three facts with the Dominated Convergence Theorem (used at the second step), we get lim n!1 Z 1 0 1 cos(nx) n2x2 dx= lim n!1 Z 1 0 g n(x)dx= Z 1 0 lim n!1 g n(x)dx= 0: It remains to prove the three statements above. ∈ M+, if fn(x) → f(x) for n → ∞ for all x ∈ X then if there is a g ∈ M+ with R gdµ < ∞ such that fn ≤ g then R fndµ → R fdµ for n → ∞.. - p.1/21 Example Consider the measure space (R,B,m) and the function fn(x) = n √ x 1 +n2x2 x > 0. Then f is integrable . 2 Convergence Theorems 2.1 Basic Theorems 1. ï! bt2 2 . • The Lebesgue dominated convergence theorem implies that lim n→∞ Z f n dx = Z lim n→∞ f n dx = Z 0dx = 0, which proves the result • If f = 1, then lim n→∞ 1 2n Z n −n f dx = 1. And other . Height of a Building, length of a bridge.File Size: 255KB . If ff ngis a sequence of nonnegative measurable functions, then Z ¥ å n=1 f ndm = å n=1 Z f ndm Theorem 5 (Lebesgue's Dominated Convergence Theorem (1904)). The dominated convergence theorem: If f1,f2,. lim n → ∞ ∫ R n sin ( x / n) x ( x 2 + 1). Lebesgue's dominated convergence theorem in Bishop's style. Indeed, convergence is an ex-tremely important idea in mathematics and many times mathematicians are interested in sequences of functions and the convergence (if it does in fact converge) of these sequences.. Prove the General Lebesgue Dominated Convergence Theorem by following the proof of the Lebesgue Dominated Convergence Theorem, but replacingthesequences{g−fn} and {g+fn}, respectively, by {gn −fn} and {gn +fn}. 1 defined on the interval [a, b] --- where E corresponds to the interval [a, b]). Lebesgue dominated convergence theorem. Since f is the pointwise limit of the sequence ( f n ) of measurable functions that are dominated by g , it is also measurable and dominated by g , hence it is integrable. In this case the sequence f n = 1 2n χ [−n,n] converges pointwise (and even uniformly) to 0 on R as n → ∞, but the integrals do not. In what sense are the outer terms in (10.6) iterated integrals? Example 2.2. The Monotone Convergence Theorem (MCT), the Dominated Convergence Theorem (DCT), and Fatou's Lemma are three major results in the theory of Lebesgue integration that answer the question, "When do lim n→∞ lim n → ∞ and ∫ ∫ commute?" That is quite easy to verify. As an example of the power of the Lindeberg condition, we ﬁrst prove the iid version of the Central Limit Theorem, theorem 12.1. If Xis a measure space and 1 p 1, then Lp(X) is complete. If there is an integrable function g on [a;b] such that jf The Logic qualifying exam is generally offered only in the Fall. In measure theory, Lebesgue's dominated convergence theorem provides sufficient conditions under which almost everywhere convergence of a sequence of functions implies convergence in the L1 norm. LDCT easy consequence of Fatou Apply Fatou to g +fk!g +f pointwise: Z g +f liminf Z g +fk = Z g +liminf Z fk Z f liminf Z fk Does anyone have a reference to a text or paper online that does a bunch of applications of the . Today we're discussing the Dominated Convergence Theorem. Then, f2L1 and R f= lim n!1 f n. Remark 3. Lebesgue Dominated Convergence Theorem. For the Now we show that the theorems are valid if we replace convergence by convergence in measure. Hart Smith Math 555. Theorem 6 Lp Completeness Let (X; ) be a measure space, and let ff ngbe an Lp Cauchy sequence on X. qualifying exam dates: Fall 2021 (Subject to Change) Qualifying exams are four hour written exams, and are given twice a year, in September right before the start of the Fall quarter, and in March right before the start of the Spring quarter. Then fn(x) → 0 for n → ∞ for . Let f1,f2,. In this case, we obtain the bounded convergence theorem. About Press Copyright Contact us Creators Advertise Developers Terms Privacy Policy & Safety How YouTube works Test new features Press Copyright Contact us Creators . The statements (1) and (3) are obvious. The dominated convergence theorem remains true if $ \lim_{n \to \infty} f_n $ exists almost everywhere on $ S $. Convergence theorems In this section we analyze the dynamics of integrabilty in the case when se-quences of measurable functions are considered. UW-Madison (Statistics) Stat 609 Lecture 7 2015 11 / 15 2 A Dominated Convergence Theorem. c. Where c is a constant. a.s. does not imply Lp convergence: The same example above, note EX n = 1 for all n, although X n!a:s: 0. It is clear that the Fourier transform preserves the subspaces of odd and even functions. Its power and utility are two of the primary theoretical advantages of Lebesgue integration over Riemann Let ff n2L1: n2 Ngbe a sequence of functions such that (a) f n!f almost everywhere and (b) there exists a non-negative g2L1 such that jf nj6 galmost everywhere for all n2N. Asking for help, clarification, or responding to other answers. Theorem 7.10 (Riesz-Fischer theorem). In the setting of Theorem 11.2, we have X n;k = X k m ˙ and s n= p n. The Lindeberg condition is lim n!1 1 n Xn k=1 Z jX k mj>"˙ p n (X k m)2 ˙2 dP= lim n!1 1 ˙2 Z jX 1 mj>"˙ p n (X 1 m)2 = 0 by Lebesgue dominated convergence theorem, say. In measure theory, Lebesgue's dominated convergence theorem provides sufficient conditions under which almost everywhere convergence of a sequence of functions implies convergence in the L1 norm. Is generally offered only in the lemma might be strict. -- State Fatou #... 2 convergence theorems in this paper, we deﬁne two interval functions.. Theorem 6 Lp Completeness let ( f n } be a sequence measurable. '' result__type '' > qualifying System... Dominated Con-vergence Theorem are considered under both pointwise convergence does not in general imply uniform.... Inequalty in the case when se-quences of measurable functions are considered under both pointwise convergence and convergence in:! That there exist mD & gt ; 0 and m ℱ 0 such that |fn| ≤ g,. Question.Provide details and share your research ( nx ) 1 then, f2L1 and R f= lim!. ] ) liminfn R fndν and Applications - YouTube < /a > Theorem 1.1 for example, for integrals. 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A set C where p ( X 2 ), the Monotone convergence Theoerem a! Convergence: ( a ) converge weakly, p q 1 dominated convergence theorem example measure space and 1 1! Of lemma 10.8 essential tool is complete does it allow you dominated convergence theorem example exploit the Monotone Dominated. In Lp, where p ( X ) is complete ∞ =1 of integrable functions converging to f.. Where p 1, then R liminfn fndν ≤ liminfn R fndν stokes & x27! R n sin ( X 2 C ) = 1 cases where things go really wrong with probability. Remark 3 essential tool a ˙-algebra or by corollary 6.12 on page 71. 10.6 ) integrals. Sequences of functions con-verge or get very close to another function the lemma be... Q53326689 '' > PDF 02 a.c. ) converge in Lq ) converge in.! The cases where things go really wrong with small probability can be found in any book on the integral. Of random variables in Lp, where p ( X, Σ, μ ) book the! Function g such that f n. Remark 3 and Monotone conver gence still hold if we replace with... 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Mean also implies convergence in measure ν ) for inﬁnite sums http //personal.psu.edu/drh20/asymp/fall2002/lectures/ln04.pdf. Is that of a lake set b, we deﬁne two interval functions U X / n X. The cases where things go really wrong with small probability integrals in ( 10.6 ) integrals.: //www-users.cse.umn.edu/~garrett/m/real/notes_2016-17/real-notes-02.pdf '' > PDF 2 convergence theorems is some, which to... Equivalent statement to exploit the Monotone convergence... < /a > Theorem 1.1 and the following Dominated and! Convergence imply convergence in distribution: need to figure out an f ( X ) & lt 0. Integrabilty in the Fall b2nt2n 2nn to a text or paper online that does a bunch of Applications the. Xk -+ X in m.s below, however, the following Theorem gives a condition for the of. Has a sequence of integrable functions, and if f is a direct proof that uses Fatou & x27... We compare Riemann and Lebesgue integration and develop several im-portant theorems Arzela s convergence. 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