A Measurable Selector in Kadison’s Carpenter’s Theorem

We show the existence of a measurable selector in Carpenter’s Theorem due to Kadison. This solves a problem posed by Jasper and the first author in an earlier work. As an application we obtain a characterization of all possible spectral functions of shift-invariant subspaces of $L^{2}(\mathbb{R}^{d})$ and Carpenter’s Theorem for type $\text{I}_{\infty }$ von Neumann algebras.

EKSISTENSI SELEKTOR TERUKUR PADA FUNGSI BERNILAI HIMPUNAN DI DALAM RUANG BANACH TAK SEPARABEL

In the Kuratowski-Ryll Nardzewski theorem, measurable selector of the set-valued functions exist if the value of set-valued function is a subset of a separable Banach spaces. In this article show that the set-valued function has a measurable selector in non separable Banach spaces.

An effective selection theorem

A recent result of J.P. Burgess [1] states:Theorem 0. Let F be a multifunction from an analytic subset T of a Polish space to a Polish space X. If F is Borel measurable, Graph(F) is coanalytic in T × X and F(t) is nonmeager in its closure for each t Є T, then F admits a Borel measurable selector.The above result unifies and significantly extends earlier results of H. Sarbadhikari [8], S.M. Srivastava [9] and G. Debs (unpublished). The reader is referred to [1] for details.The aim of this article is to give an effective version of Theorem 0. We do this by proving a basis theorem for Π11 sets which are nonmeager in their closure and satisfy a local version of the measurability condition in Theorem 0. Our basis theorem generalizes a well-known result of P.G. Hinman [4] and S.K. Thomason [10] (see also [5] and [7, 4F.20]). Our methods are similar to those used by A. Louveau to prove that a , σ-compact set is contained in a , σ-compact set (see [7, 4F.18]).The paper is organized as follows. §2 is devoted to preliminaries. In §3, we prove the basis theorem and deduce as a consequence an effective version of Theorem 0. We show in §4 how our methods can be used to give alternative proofs of some known results.Discussions with R. Barua, B.V. Rao and V.V. Srivatsa are gratefully acknowledged. I am indebted to J.P. Burgess for drawing my attention to an error in an earlier draft of this paper.

Multifunctions of Souslin type

LetSandXbe any two sets; then a mapping Γ which assigns to each pointtinSa set Γ(t) of points inXis called amultifunctionfromSintoX. Aselectorfor Γ is a functionffromSintoXsuch thatf(t)∈ Γ(t) for eacht. We introduce here a class of multifunctions which is both well-supplied with measurable selectors and yet is comprehensive enough to include those kinds of multifunction which have been most commonly studied before. Hence in order to show that a multifunction with non-empty values, which may arise naturally in an implicit function problem, has a measurable selector, it is sufficient to show that it is of Souslin type.

Measurable selections in normed spaces

It is sometimes desirable to know in what circumstances a measurable set valued function admits a measurable selector; this problem occurs regularly in the theory of optimal control (see for example (3) and (7)). In this paper we demonstrate the existence of measurable selectors in two particular cases where the choice of selector has a simple geometrical interpretation, namely that of being a “ nearest-point ” selector, as is explained in detail below. This work derives in part from that of C. Castaing, particularly from Théorème 3.4 of (2), of which this is an extension.

Some Selection Theorems for Measurable Functions

Let F: X → Y be a multifunction from X to Y. Then, given measure-theoretic or topological structures on X and Y, it is possible in various ways to define the measurability of F. The selection problem is to determine which structures on X and Y and which definitions of measurability of F ensure that F will have a measurable selector. This problem has been studied recently in papers by Castaing (2) and Kuratowski and Ryll-Nardzewski (6). In the latter paper, the problem is studied for its own interest. The former uses solutions of the problem to obtain general Filippov-type theorems. (See, for example, the corollaries following Theorems 2 and 3 of Castaing's paper.) For other material on Filippov's results see, among others, (3; 4; 5; 7; 9).