Cramér–Wold Uniqueness for Infinite Signed Measures

Definitions and mathematical statement

The source asks for a Cramér--Wold-type uniqueness theorem beyond finite positive measures, allowing signed measures with singular or infinite mass near the origin. For a finite measure, halfspace data may be written \[ H(u,t)=\{x\in{\mathbb{R}}^d:\langle u,x\rangle>t\}, \qquad u\in S^{d-1},\ t\in{\mathbb{R}}, \] with a fixed open/closed boundary convention. The problem is to specify an admissible class and halfspace transform for which equality of these data determines the measure.

A natural framework, not supplied as a hypothesis in the record, is signed Radon measures on ${\mathbb{R}}^d\setminus\{0\}$, with finite total variation on sets bounded away from zero and suitable control at infinity. For example, the stronger tail condition \[ |\mu|(\{x:\|x\|\geq\varepsilon\})<\infty\qquad(\varepsilon>0) \] would make $\mu(H(u,t))$ finite for $t>0$. With a chosen class $\mathcal C$ and these admissible halfspaces, the desired injectivity statement would be \[ \mu,\nu\in\mathcal C,\quad \mu(H(u,t))=\nu(H(u,t))\ \forall u\in S^{d-1},\ t>0 \quad\Longrightarrow\quad\mu=\nu. \] The source does not assert that the example tail condition alone suffices; further growth, integrability, or structural hypotheses are what the question asks to determine.

If measures are instead defined on all of ${\mathbb{R}}^d$, halfspaces staying away from the origin cannot detect an atom at zero, so that atom must be excluded, specified, or measured by additional data. Halfspaces meeting an infinite-mass singularity also may not have finite signed values: unrestricted subtraction of infinite positive and negative masses is undefined. A principal-value or distributional transform would require its own definition and cannot be inserted silently. Local finiteness on the punctured space alone also does not control unbounded-halfspace tails.

The exact measure domain, sign and variation conventions, admissible halfspaces, near-zero and infinity bounds, and any regularization are absent. The supplied assessment checked 2026-08-21 calls the broad infinite-signed-measure/Radon-transform extension unresolved in the extracted generality. This is a request for a well-defined injectivity class, not an unqualified assertion that all infinite signed measures are determined by possibly undefined halfspace masses.

Short English statement

Find hypotheses making halfspace data determine signed measures that may diverge near zero. The domain, finite-variation tails, admissible halfspaces, and treatment of an origin atom or undefined infinite signed values must be fixed.

Sources

• Local UnsolvedMath registry, permanent ID 30000708, catalogue entry, Cramér--Wold Uniqueness for Infinite Signed Measures.
• Recent Developments in Financial and Insurance Mathematics and the Interplay with the Industry (2007), Oberwolfach Reports, pp. 549--590, DOI https://doi.org/10.4171/owr/2007/10.
• The supplied statement names the Cramér--Wold theorem as background, without a separate bibliographic entry. Supplied literature triage checked 2026-08-21 cites only the report.