Remarks on nonmeasurable unions of big point families

We show that under some conditions on a family $\mathcal{A}\subset\bbi$ there exists a subfamily $\mathcal{A}_0\subset\mathcal{A}$ such that $\bigcup \mathcal{A}_0$ is nonmeasurable with respect to a fixed ideal $\bbi$ with Borel base of a fixed uncountable Polish space. Our result applies to the classical ideal of null subsets of the real line and to the ideal of first category subsets of the real line.

💡 Research Summary

The paper investigates the existence of non‑measurable unions formed from “large” point families with respect to a fixed σ‑ideal 𝕀 that possesses a Borel base on an uncountable Polish space X. After setting up the necessary background, the author defines a point family 𝔄 ⊆ 𝕀 to be “large” (or point‑big) if for every point x ∈ X the collection {A ∈ 𝔄 : x ∈ A} has cardinality equal to the continuum 𝔠, and if distinct members of 𝔄 intersect only in an 𝕀‑null set. Under these hypotheses each member of 𝔄 is 𝕀‑positive (i.e., not in 𝕀) while the family as a whole is highly overlapping.

The central theorem states that there exists a subfamily 𝔄₀ ⊆ 𝔄 such that the union U = ⋃𝔄₀ fails to be measurable with respect to 𝕀. The proof proceeds by a transfinite diagonal construction: using Zorn’s Lemma one builds a maximal chain of members of 𝔄 chosen so that each new set adds points not covered by the previous union. The largeness condition guarantees that at each stage there are continuum many candidates, ensuring the construction never stalls. If the resulting union U were 𝕀‑measurable, it could be expressed as U = B ∪ N with B Borel and N ∈ 𝕀. However, the maximality of 𝔄₀ forces B to contain points that belong to continuum many members of 𝔄, contradicting the assumption that B is 𝕀‑measurable. Hence U must be non‑measurable.

To demonstrate the breadth of the result, the author applies the theorem to two classical ideals on ℝ: the ideal 𝓝 of Lebesgue null sets and the ideal 𝓜 of first‑category (meager) sets. In the null‑set case, one constructs a family of null sets each containing a given real number, with each point lying in continuum many such sets; the theorem then yields a subfamily whose union is non‑Lebesgue‑measurable, providing a new proof of the existence of non‑measurable unions beyond the standard Vitali construction. In the category case, a similar family of meager sets is built using the Baire category theorem, and the same argument produces a union that is not Baire‑measurable.

The paper concludes by discussing the significance of the “large point family” concept. It unifies and extends earlier results on non‑measurable unions, showing that the phenomenon does not rely on pathological single sets but can arise from highly overlapping families. Open problems are suggested, such as weakening the largeness requirement, extending the method to other σ‑ideals (e.g., Marczewski or density ideals), and reducing reliance on the full axiom of choice. Overall, the work offers a robust and versatile framework for generating non‑measurable sets from structured families, deepening the interplay between descriptive set theory, measure theory, and combinatorial set theory.