Baire Category of Purely Singular Copulas

Definitions and mathematical statement

For a positive integer $d$, let $\lambda_d$ denote Lebesgue measure on $[0,1]^d$. A $d$-dimensional copula is a function $C:[0,1]^d\to[0,1]$ of the form \[ C(u_1,\ldots,u_d)=\mu_C\!\left(\prod_{j=1}^d[0,u_j]\right), \] where $\mu_C$ is a Borel probability measure on $[0,1]^d$ whose every one-dimensional marginal is uniform on $[0,1]$. In particular, $C(u)=0$ whenever a coordinate of $u$ is zero, and $C(1,\ldots,1,t,1,\ldots,1)=t$ for every coordinate position and $t\in[0,1]$. Write $\mathcal C_d$ for the set of these copulas. A copula is purely singular if $\mu_C\perp\lambda_d$, meaning that there is a Borel set $N\subseteq[0,1]^d$ with $\lambda_d(N)=0$ and $\mu_C(N)=1$. Set \[ \mathcal S_d=\{C\in\mathcal C_d:\mu_C\perp\lambda_d\}. \] The supplied expanded statement defines \[ D_1(C_1,C_2)=\int_{[0,1]^d}|C_1(u)-C_2(u)|\,\mathrm du. \] A subset of $\mathcal C_d$ is meager in $(\mathcal C_d,D_1)$ if it is a countable union of sets whose closures have empty interior in that ambient metric space. A nonmeager subset is said to be of second category.

Question as expanded in the supplied record. Is $\mathcal S_d$ nonmeager in $(\mathcal C_d,D_1)$? Equivalently, is it impossible to write $\mathcal S_d$ as a countable union of nowhere dense subsets of this ambient space?

Unresolved qualifications. The original quotation asks whether purely singular copulas are of second category in $(\mathcal C_d,D_1)$, but does not itself define $D_1$. The integral formula above comes from the supplied expanded statement; its agreement with the metric intended in the cited report requires verification. The accompanying literature assessment distinguishes a uniform metric from $D_1$, so the intended metric and that assessment must be reconciled before this draft is treated as a faithful source formalization. The record also does not specify the intended range or quantification of $d$; the question is therefore displayed with $d$ as a parameter, without asserting that the original problem concerns every positive integer or that it excludes $d=1$.

Short English statement

The supplied question asks whether purely singular copulas form a nonmeager subset of the space of copulas under the displayed integral distance. Pure singularity concerns the associated probability measure. The intended metric and dimension range require source verification. The record labels the problem open; that label and its dated literature assessment are supplied curation, not independently verified conclusions.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30002859$, exact source problem code catalogue entry, title Baire Category of Purely Singular Copulas, in the collection Oberwolfach Reports Open Problems. The record attributes its provenance to mathematical workshop and open-problem discussions documented in Oberwolfach Reports. No individual proposer is supplied; the proposed year is recorded as 2015.
• Supplied report citation: Copulae: On the Crossroads of Mathematics and Economics (2016), Oberwolfach Reports, DOI 10.4171/owr/2015/20, pp. 1085--1137. Supplied source URL and literature-source link labeled Oberwolfach Report 20: https://doi.org/10.4171/owr/2015/20. The supplied title year 2016 is retained alongside the DOI containing 2015; their bibliographic relationship has not been verified.
• Supplied literature-source link: A typical copula is singular, https://www.sciencedirect.com/science/article/pii/S0022247X15004527. No authors or further bibliographic details are supplied.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open and states that purely singular copulas are typical in a uniform metric while the corresponding second-category question in the $D_1$ metric remains open. This is supplied dated curation, described as literature triage derived from the supplied OWR curation, and is not independently verified here. Its time-sensitive status, cited claims, and compatibility with the displayed definition of $D_1$ require independent verification.
• The registry labels the statement corrected_verified and says that mathematical content was preserved while notation, typography, or omitted definitions were repaired for publication. These are supplied editorial assertions, not mathematical verification of this draft or confirmation that the expanded metric definition matches the report.