Polynomial Product Weights, Ideal Comparison, and Weighted Wijsman Convergence for Triple Sequences of Closed Sets

Abstract

This paper studies weighted Wijsman ideal convergence for triple sequences of closed sets in Menger probabilistic metric spaces with polynomial product weights \(w(i,j,k)=i^{\alpha}j^{\beta}k^{\gamma}\). For this class of weights, we compute the weighted density of cylindrical, planar, and diagonal index sets and prove that different exponent triples yield distinct ideals. The map \((\alpha,\beta,\gamma)\mapsto\mathcal{I}_{w}\) is an injective order-reversing embedding into the lattice of ideals on \(\mathbb{N}^{3}\). Under separability, property~(AP3), and uniform equi-Wijsman-regularity, we prove that \(\mathcal{I}_{w}^{\psi}\)-convergence is equivalent to \(\mathcal{I}_{w}^{*\psi}\)-convergence. In the metric-induced Menger model, the regularity hypothesis holds for all nonempty closed sets, so the equivalence theorem applies directly in Euclidean spaces.