On repeated sequential closures of constructible functions in valuations

TL;DR

This paper proves that repeatedly taking sequential closures of constructible functions eventually yields the entire space of generalized valuations, which is important for applications in valuation theory on manifolds.

Contribution

It establishes that infinitely iterated sequential closures of constructible functions equal the space of generalized valuations, strengthening previous density results.

Findings

Sequential closures of constructible functions become dense in generalized valuations after infinitely many iterations.

The result supports applications in valuation theory on manifolds.

Provides a foundational property for future research in valuations on manifolds.

Abstract

The space of constructible functions form a dense subspace of the space of generalized valuations. In this note we prove a somewhat stronger property that the sequential closure, taken sufficiently many (in fact, infinitely many) times, of the former space is equal to the latter one. This stronger property is necessary for some applications in the theory of valuations on manifolds.