Measure, Compactification and Representation

Abstract

The theory of measure on topological spaces has in recent years found its most natural setting in the study of pavings and measures on such pavings (see e.g. [1-3; 5; 6; 10; 19; 22; 32; 33]. In this setting the relationship between measure and topology crystallizes since one concentrates primarily on the simpler internal lattice structure associated with sublattices of the topology rather than on the more complex topological structure.

Measure theoretic techniques in topology and mappings of replete and measure replete spaces

“…. This is a fairly easy consequence of the Portmanteau Theorem (see [26] for details). We also note that if L is separating and disjunctive as well as delta normal, then the vague topology on X ,when X is identified with the collection of meas-ures concentrated at a point, coincides with the topology on X having as a base for the closed sets the lattice L , that is, the T ( L ) topology.…”

“…We also note that if L is separating and disjunctive as well as delta normal, then the vague topology on X ,when X is identified with the collection of meas-ures concentrated at a point, coincides with the topology on X having as a base for the closed sets the lattice L , that is, the T ( L ) topology. (Again see [26].) If X is a Tychonoff space we denote by 6X the Stone-Cech compactification of X , and by \)X , the real compactification of X .…”

Hausdorff compactifications and zero-one measures II

“…Recall that (see, e.g., [1,3,4,14]) a measure on a Boolean algebra A is a non-negative real-valued function µ on A such that µ(a ∨ b) = µ(a) + µ(b) for all a, b ∈ A with a∧b = 0; in the case when µ(A) = {0, 1}, µ is called a zero-one measure.…”

“…It was well known (see [1,3,4,14]) that, when C is a normal base of X, then the space I R (C) (of all regular zero-one measures on the Boolean subalgebra b(C) of the Boolean algebra exp(X) (of all subsets of X, with the natural operations), generated by the sublattice C of exp(X)) is a Hausdorff compactification of X equivalent to ω(X, C) and max(C ). The second problem was: Problem 1.2.…”

A general measure decomposition theorem by means of the generalized wallman remainder

“…It is known that the topological space (IR(t), tW(t)) is compact and T x ; it is T 2 if and only if £ is normal. (See, for example, [2] and [13].) Consider the map tf>: X ->//?…”