Radial continuous valuations on star bodies and star sets

TL;DR

This paper demonstrates that radial continuous valuations on star bodies can be uniquely extended to bounded star sets and provides an integral representation, including a decomposition into positive valuations.

Contribution

It introduces a unique extension of radial continuous valuations to bounded star sets and offers an integral representation based on the radial function.

Findings

Unique extension of valuations to bounded star sets

Integral representation of valuations on dense subsets

Decomposition into positive radial valuations

Abstract

We show that a radial continuous valuation defined on the $n$-dimensional star bodies extends uniquely to a continuous valuation on the $n$-dimensional bounded star sets. Moreover, we provide an integral representation of every such valuation, in terms of the radial function, which is valid on the dense subset of the simple Borel star sets. We also show that every radial continuous valuation defined on the $n$-dimensional star bodies can be decomposed as a sum $V=V^+-V^-$, where both $V^+$ and $V^-$ are positive radial continuous valuations.