The quasi topology associated with a countably subadditive set function

Abstract

This is a general study of an increasing, countably subadditive set function, called a capacity, and defined on the subsets of a topological space <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>X</mml:mi> </mml:math> . The principal aim is the study of the “quasi-topological” properties of subsets of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>X</mml:mi> </mml:math> , or of numerical functions on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>X</mml:mi> </mml:math> , with respect to such a capacity <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>C</mml:mi> </mml:math> . Analogues are obtained to various important properties of the fine topology in potential theory, notably the quasi Lindelöf principle (Doob), the existence of a fine support (Getoor), and the theorem on capacity for decreasing families of sets (Brelot). This analogy becomes an actual identity if a certain compatibility is assumed betweeh the capacity <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>C</mml:mi> </mml:math> and a new homology (called “fine”) on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>X</mml:mi> </mml:math> . Sufficient conditions are obtained with a convex cone of lower semicontinuous functions on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>X</mml:mi> </mml:math> .