The best available definition of a subset of an infinite dimensional, complete, metric vector space being ``small' is Christensen's Haar zero sets, equivalently, Hunt, Sauer, and Yorke's shy sets. The complement of a shy set is a prevalent set. There is a gap between prevalence and likelihood. For any probability on , there is a shy set with . Further, when is locally convex, any i.i.d. sequence with law repeatedly visits neighborhoods of only a shy set of points if the neighborhoods shrink to at any rate.
Given a topological system on a -compact Hausdorff space and its factor we show the existence of a largest topological factor containing such that for each -invariant measure , . When a relative variational principle holds, .
A (discrete) group is said to be maximally almost periodic if the points of are distinguished by homomorphisms into compact Hausdorff groups. A Hausdorff topology on a group is totally bounded if whenever there is such that . For purposes of this abstract, a family with a totally bounded topological group is a strongly extraresolvable family if (a) \vert G\vert$">, (b) each is dense in , and (c) distinct satisfy ; a totally bounded topological group with such a family is a strongly extraresolvable topological group.
We give two theorems, the second generalizing the first.
Theorem 1. Every infinite totally bounded group contains a dense strongly extraresolvable subgroup.
Corollary. In its largest totally bounded group topology, every infinite Abelian group is strongly extraresolvable.
Theorem 2. Let be maximally almost periodic. Then there are a subgroup of and a family such that
(i) is dense in every totally bounded group topology on ;
(ii) the family is a strongly extraresolvable family for every totally bounded group topology on such that ; and
(iii) admits a totally bounded group topology as in (ii).
Remark. In certain cases, for example when is Abelian, one must in Theorem 2 choose . In certain other cases, for example when the largest totally bounded group topology on is compact, the choice is impossible.
A Lie subalgebra of is said to be finitary if it consists of elements of finite rank. We show that, if acts irreducibly on , and if is infinite-dimensional, then every non-trivial ascendant Lie subalgebra of acts irreducibly on too. When , it follows that the locally solvable radical of such is trivial. In general, locally solvable finitary Lie algebras over fields of characteristic are hyperabelian.
This paper is devoted to a study of multivariate nonhomogeneous refinement equations of the form where is the unknown, is a given vector of functions on , is an dilation matrix, and is a finitely supported refinement mask such that each is an (complex) matrix. Let be an initial vector in . The corresponding cascade algorithm is given by In this paper we give a complete characterization for the -convergence of the cascade algorithm in terms of the refinement mask , the nonhomogeneous term , and the initial vector of functions .
We prove that a Banach space has the compact range property (CRP) if and only if, for any given -algebra , every absolutely summing operator from into is compact. Related results for -summing operators () are also discussed as well as operators on non-commutative -spaces and -summing operators.