The Radon-Nikodym theorem for random measures

Translated from Ukrainskii Materaaticheskii Zhurnal, Vol. 41, No. 1, pp. 63–67, January, 1989.     

The Radon-Nikodym theorem for von neumann algebras

Let ϕ be a faithful normal semi-finite weight on a von Neumann algebraM. For each normal semi-finite weight ϕ onM, invariant under the modular automorphism group Σ of ϕ, there is a unique self-adjoint positive operatorh, affiliated with the sub-algebra of fixed-points for Σ, such that ϕ=ϕ(h·). Conversely, each suchh determines a Σ-invariant normal semi-finite weight. An easy application of this non-commutative Radon-Nikodym theorem yields the result thatM is semi-finite if and only if Σ consists of inner automorphisms. Partially supported by NSF Grant # 28976 X. Partially supported by NSF Grant # GP-28737 This revised version was published online in November 2006 with corrections to the Cover Date.     

Representation theorems of the dual of Lebesgue-Bochner function spaces

A model for binary evolution is introduced which can determine whether the rotation of components is synchronized with the orbital motion, and can calculate the evolution of both the synchronized and non-synchronized binary systems. With this model, the evolution of a binary system consisting of a 9 M_⊙ star and a 6 M_⊙ star is studied with mass transfer Case B. The result shows that the synchronization of the rotational and orbital periods can be reached when the binary system is a detached system and before the occurrence of the first mass transfer. After the onset of the first mass transfer, the binary system becomes non-synchronized. The mass accepted component (the secondary) rotates faster with a period much smaller than that of the orbital motion.     

A Radon-Nikodym type theorem for functionals

Let ($\text{Ω, τ, m}$) be a finite, nonatomic, separable measure space. This paper extends the Radon-Nikodym theorem to odd, disjointly additive, m-continuous functionals whose domain consists of all differences of characteristic functions which belong to a given subspace of L^∞(m). Such a functional will possess a density in L¹(m) provided that the subspace is weak¹-closed and separates sets; the conclusion can fail if the latter hypothesis is removed. Analogous results are obtained for functionals which are not necessarily odd.     

An abstract Radon-Nikodym theorem

In this paper we obtain a Radon-Nikodym theorem for positive linear functionals on a B¹-algebra M. Some corollaries analogous to those obtained in the classical case are also obtained here. It is known that if X is a Banach space, then the space $\text{L}{}_1\text{(Ω, X)}$ of Bochner integrable functions on a probability space Ω with values in X is the completion (in a suitable topology) of the tensor product $\text{L}{}_1\text{(Ω) ? X}$. Using our theorem, it is possible to extend this result for certain linear mappings from M ? X to X.     

A unifying Radon-Nikodym theorem for vector measures

It is a known fact that certain derivation bases from martingales with a directed index set. On the other hand it is also true that the strong convergence of certain abstract martingales is a consequence of the Radon-Nikodym theory for vector measures (cf. Uhl, J. J., Jr., Trans. Amer. Math. Soc.145 1969, 271–285). Many other connections and applications of the latter theory with multidimensional problems in stochastic processes and representation theory are known (cf. Dinculeanu, N., Studia Math.25 1965, 181–205; Dinculeanu, N., and Foias, C., Canad. J. Math.13 1961, 529–556; Rao, M. M., Ann. Mat. pura et applicata76 1967, 107–132; Rybakov, V. I., Izv. Vys?. U?ebn. Zaved. Matematika19 1968, 92–101; Rybakov, V. I., Dokl. Akad. Nauk SSSR180 1968, 620–623). Starting from various vantage points, many authors have proposed several hypotheses for establishing abstract Radon-Nikodym theorems. In view of the great interest and importance of this problem in the areas mentioned above, it is natural to obtain a unifying result with a general enough hypothesis to deduce the various forms of the Radon-Nikodym theorem for vector measures. This should illuminate the Radon-Nikodym theory for vector measures and stimulate further work in abstract martingale problems. In this paper the first problem is attacked, leaving the martingale part and other applications for another study.The main result (Theorem 7 of Section 2) provides the desired unification and from if the Dunford-Pettis theorem, the Phillips theorem and several others are obtained. As martingale-type arguments are constantly present, a careful reader may note the easy translation of the hypothesis to the martingale convergence problem but we treat only the Radon-Nikodym problem using the language of measure theory and linear analysis.     

An approximate functional Radon-Nikodym theorem

We prove that for two linear and positive functionals (not necessarily Daniell)J andI on a lattice unitary algebraB of functions such thatJ is absolutely continuous with respect toI, one can expressJ as follows: , where (v _m)m is a fixed sequence inB, for allf inB. This result is the “functional” similar of a previous deep result due to C. Fefferman. The comments and the counterexamples which we are introducing show that the main result (i.e sequential approximation) cannot be improved.     

A Nagata-like theorem for certain function spaces

Let X be a space, and let A be a zero-dimensional topological ring. In this paper we will consider a few natural questions that arise when studying the space C _p (X, A), the ring of continuous functions from X to A, endowed with the topology of pointwise convergence. It will be shown that the zero-dimensionality of the codomain plays a vital role in this study. An upper and lower bound will be determined for the density of C _p (X, A) using the density of A and the weight of X. The character of C _p (X, A) will be computed, thus characterizing when C _p (X, A) is metrizable. Lastly, we will consider the topological dual space of C _p (X, A) and use it to prove a Nagata-like theorem.     

The duality between Asplund spaces and spaces with the Radon-Nikodym property

A Banach spaceX is an Asplund space (a strong differentiability space) if and only ifX ^* has the Radon-Nikodym property.     

Radon-Nikodym property for the projective tensor product of Köthe function spaces

Let E be an order continuous Köthe function space (or an order continuous Banach lattice) and X be a dual Banach space. Then , the projective tensor product of E and X, has the Radon-Nikodym property if and only if both E and X do.     

The Levin-Golovin theorem for sobolev spaces

We obtain sufficient conditions for the basis property of the family of exponentials