Measures Determined by the Restriction of Convolution Powers to the Proper Concave Cone

TL;DR

This paper proves that measures on a convex cone are equal if their convolution powers agree on a proper concave cone, extending previous results and providing counterexamples in two dimensions.

Contribution

It establishes new conditions under which measures are uniquely determined by their convolution powers on specific cones, generalizing prior work.

Findings

Measures are equal if their convolution powers agree on a proper concave cone.

Counterexample shows agreement on a half-plane does not imply measure equality.

Extension of measure uniqueness results to convex cones in R^n.

Abstract

Let $\mu$ and $\nu$ be two non-degenerate finite signed Borel measures defined on a proper convex cone of $\mathbb{R}^n$. We prove that if all convolution powers of $\mu$ and $\nu$ are appropriately equal (and non-zero) on a proper concave cone of $\mathbb{R}^n$, the measures are equal. A similar but more general result for measures defined on $\mathbb{R}$ can be found in [2]. We also provide an example of two-dimensional measures, which indicates that equality of measures and their appropriate convolution powers on a half-plane is not enough for equality of measures.