Theorem 4.1.8 (Lebesgue分解定理) 若μ,ν 为σ-有限测度,则存在测度 νr,νs 与可测函数 g 使得ν=νr+νs . 其中 νs⊥μ, νr(E)=∫Egdμ . proof : 由于可以分解全空间,不妨设 μ,ν 为有限测度. 取G={g:g≥0,∀E,∫Egdμ≤ν(E)} . ...
假设\left\lbrace \nu_j\right\rbrace是一列正测度,若\forall j,\nu_j\bot \mu则\sum \nu_j\bot \mu;若\forall j,\nu_j \ll\mu,则\sum \nu_j\ll \mu. Proof: (1).对于\nu_j\bot \mu,不妨设\nu_j集中在A_j,\mu集中在B_j,即X=A_j\cup B_j,\nu_j(B_j)=\mu(A_j)=0.令...
Yet another proof of the Radon-Nikodym theorem, this MONTHLY - Selke - 2002 () Citation Context ..., analysis, or probability theory. Usually they use the Hahn decomposition theorem for signed measures, the Riesz representation theorem for functionals on Hilbert spaces, or martingale theory, ( ...
Theorem 2.34. The theorem is valid without assuming F and Z separable; but the proof involves the lifting theory (Dinculeanu (1967, Theorem 5, p. 269) and A. and C. Ionescu Tulcea (1969)). The following theorem is in a certain sense the converse of the preceding one. It is used ...
Theorem 1 L et (X, S, Λ) be a f inite m easu re sp ace and f be a Λ2integ rable f unction, then 0 ( 3 f) is L ( 3 Λ ) 2integ rable and 0 3 fd 3 Λ= 0 ( 3 f ) d ( L ( 3 Λ ) ). Proof W ithout lo ss of generality, assume that f is a non2ne...
笔者在本文讨论应用Riesz-Frechet定理证明R-N定理.拟定的证明改进了文[1]中给出的简略证明,且在问题的处理方面与文[1]的论述略有不同.R-N定理的两种证明方式各有其独特之处.这两种证法都值得研究. 著录项 来源 《重庆交通大学学报:自然科学版》 |2005年第5期|167-168|共2页 作者...
In this paper, we introduce and study the signed fuzzy-valued measure and using the representation theorem of fuzzy numbers, we establish the Radon-Nikodym theorem for fuzzy-valued measurable functions with respect to fuzzy-valued measures.关键词: Fuzzy number Signed fuzzy-valued measure Fuzzy-valued...
Radon-Nikodym theoremA simplified proof is given of Doob's result that a balayage ordered collection of probability measures on a compact Hausdorff space yields a -valued supermartingale with the measures as marginal distributions. The proof shows further connections with martingale convergence theory.S...
Radon-Nikodym theoremA simplified proof is given of Doob's result that a balayage ordered collection of probability measures on a compact Hausdorff space yields a -valued supermartingale with the measures as marginal distributions. The proof shows further connections with martingale convergence theory....
Radon-Nikodym theoremA simplified proof is given of Doob's result that a balayage ordered collection of probability measures on a compact Hausdorff space yields a -valued supermartingale with the measures as marginal distributions. The proof shows further connections with martingale convergence theory....