Fatou's Lemma (法图引理) 若{fn}∞n=1{fn}n=1∞为一个由非负且可测的函数构成的序列,且对于任意n∈Nn∈N,fnfn定义在E∈ME∈M上(MM代表所有勒贝格可测集构成的集类),那么: liminfn→∞∫Efndm≥∫Eliminfn→∞fndmlim infn→∞∫Efndm≥∫Elim infn→∞fndm Proof. (Fatou's Lemma) 设:gn...
Proof: First, limn→∞∫Xfn≤∫Xf holds from the fact that fn is increasing. Second, by Fatou's lemma ∫Xfdμ≤lim∫Xfndμ .Dominated Convergence Theorem: If \{ f_n: X \to [0,\infty] \} is a sequence of measurable functions with f_n \to f almost everywhere, and if there ...
Lemma 1 (Fatou's lemma) For a sequence of measurable non-negative functions {fn} , ∫lim inffndμ≤liminf∫fndμ. Proof: By definition of inf , infk≥nfk≤fk for all k≥n , from which follows ∫infk≥nfk≤infk≥n∫fk.(1) ...
S. Simons. An eigenvector proof of Fatou's lemma for continuous functions. Math. Intelligencer, 17:67 - 70, 1995. Universit´e d'Orl´eans, BP 6759, F-45067 Orl´eans Cedex 2, France E-mail address: hermann.pfitzner@univ-orleans.fr...
(2.10) Fatou’s Lemma and Lebesgue’s Convergence Theorem for Measures 143 Proof: (a) As f is nonnegative and lower semi-continuous, there exists an increas- ing sequence of nonnegative continuous bounded functions vk on X such that vk(x)T f(x) Vx E X. Similarly, as each vk is a...
,(Q))’ 其中c=c(n:m,ro,九). r,妒(一)≤Xn≤妒(一)+2M_1r).记 局部BMOI区域G=((一,Xn),I√一屹l≤2M一1 理4.3的条件.构造区域Q={∥,粕):Ⅳ一蜴I≤2M—r.妒∥)≤Xn≤y(一)),则 CQc B(Q,百3朋1r)NDB(Q,寻肘一1r)nD. 由定理3.4和【5,Lemma2.2】知Dirichlet问题(3.5)在区域...
A unifying note on Fatou’s lemma in several dimensions,”Math - BALDER - 1984 () Citation Context ...rt of (Ω,Σ, μ). Proposition 2.3. In Theorem 2.2 {fnj}nj can always be chosen in such a way that for some measurable function f∗ : Ωpa → E ‖fnj(ω))− f∗(ω...
The Fatou lemma approach to the existence in quasilinear elliptic equations with natural growth termsdoi:10.1080/17476930903276241In this article, we give a new proof of the existence of bounded solutions for the problem using the method introduced in Boccardo et al. [Existence de solutions non ...
A general Fatou Lemma is established for a sequence of Gelfand integrable functions from a vector Loeb space to the dual of a separable Banach space or, with a weaker assumption on the sequence, a Banach lattice. A corollary sharpens previous results in the finite-dimensional setting even for ...