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...
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...
fatous lemma and lebesgues convergence theorem for measures费托引理和勒贝格收敛定理的措施 Journal of Applied Mathematics and Stochastic Analysis, 13:2 (2000), 137-146. FATOU’S LEMMA AND LEBESGUE’S CONVERGENCE THEOREM FOR MEASURES ONISIMO HERNiNDEZ-LERMA CINVESTAV-IPN, Departamento de Matembticas...
A Green proof of Fatou’s theorem - O’Neill () Citation Context ...sure on R. The next lemma is also standard, its proof being accomplished by a straightforward application of Hall’s lemma and the strong Markov property. A complete probabilistic argument is given in =-=[14]-=- so ...
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...
differential equations, the equality follows.%Then the proof of Theorem 1 proceeds as before, but now it must be noted that Γ(x,t;·,t0)→Γ(x,t;·,0) strongly in as t0→ 0.%One must make a similar adjustment to Lemmas 4′, 5′, and Theorem 3, using Chabrowski's Theorem 1,...