Measure representation of locally finite sigma-continuous valuations

Prove that every locally finite sigma-continuous valuation on the family of compact convex subsets of \(\mathbb{R}^d\) admits a representation \(\varphi(K)=\mu(\{F:F\cap K\neq\varnothing\})\) for all compact convex \(K\), where \(\mu\) is a locally finite signed measure on the family of closed convex subsets of \(\mathbb{R}^d\).

Background

The paper studies valuations on compact convex sets and notes that their finite additivity does not generally yield a Jordan decomposition or a straightforward measure representation. A locally finite valuation is one whose absolute values remain bounded on all convex bodies contained in any fixed compact set, while sigma-continuity means continuity along decreasing sequences of compact convex sets.

The conjecture seeks to characterize all locally finite sigma-continuous deterministic valuations through measures on the space of closed convex sets. The paper proves a random analogue under substantially stronger assumptions—non-negativity, infinite divisibility, sigma-continuity, and independent increments—but does not establish the deterministic conjecture in full generality.