Borel Measure Extensions of Measures Defined on Sub-σ-Algebras

Abstract

We develop a new approach to the measure extension problem, based on nonstandard analysis. The class of thick topological spaces, which includes all locally compact and all K-analytic spaces, is introduced in this paper, and measure extension results of the following type are obtained: If (X, T) is a regular, Lindelo f, and thick space, A/_[T] is a _-algebra, and & is a finite measure on A, inner regular with respect to the closed sets in A, then & has a Radon extension. The methods developed here allow us to improve on previously known extension results. 2000Academic Press

When do type structures contain all hierarchies of beliefs?

“…In general, the answer is no. (See Aldaz and Render (2000) for examples.) But when is an analytic subset of a Polish space (and, in particular, Polish spaces are analytic subsets of Polish spaces), any probability measure on the sub-σ -algebra can be extended to a probability measure on the Borel sets.…”

“…(See, also, Landers and Rogge (1974) and Yershov (1974). Aldaz and Render (2000) provide other topological conditions. )…”

Topological Sigma-Semiring Separation and Ordered Measures in Noetherian Hyperconvexes

“…The developments in topological measure theory are propelled by Alexandrov and Varadarajan, considering that the topological spaces are always completely regular as well as Hausdorff [6,7]. The fundamental question in measure theory and its topological variants is the extensibility of σ−algebras [8]. The approach of Alexandrov is based upon the finitely additive set-valued functions in a topological space, and the approach of Varadarajan is primarily based upon the C b algebraic forms of bounded continuous real-valued functions in the completely regular spaces.…”

Pushing down infinite Loeb measures

“…The question of when st is measurable has been extensively studied; see [4] and [9] for early papers on the question, and the discussion following Theorem 3.2 of [16] for more recent results. The reader is also referred to [8], [2], [11], [12], as well as to [5] and [6] for a functional approach.…”