[Paper Review] Partitions into thin sets and forgotten theorems of Kunugi and Lusin-Novikov

E. Grzegorek, Iwo Labuda|arXiv (Cornell University)|May 17, 2017

Advanced Topology and Set Theory3 references4 citations

TL;DR

This paper uncovers and analyzes two overlooked 1930s results—by Kunugi and Lusin-Novikov—that predate and anticipate key developments in the theory of Baire category and measurable functions. It demonstrates that a function from a metric space to a topological space with the Baire property is continuous except on a first category set, even without requiring separability of the codomain, thereby resolving a foundational question posed by Kuratowski in 1935.

ABSTRACT

Let $f$ be a function from a metric space $Y$ to a separable metric space $X$. If $f$ has the Baire property, then it is continuous apart a 1st category set. In 1935, Kuratowski asked whether the separability requirement could be lifted. A full scale attack on the problem took place in the late seventies and early eighties. What was not known then, and what remains virtually unknown today, is the fact that the first impressive attempt to solve the Kuratowski problem, due to Kinjiro Kunugi and based on a theorem of Lusin and Novikov, took place already in 1936. Lusin's remarkable 1934 Comptes Rendus note soon forgotten, remained unnoticed to this day. We analyze both papers and bring the results to full light.

Motivation & Objective

• To recover and re-establish the significance of Kinjiro Kunugi’s 1936 work, which predated and anticipated later developments in the theory of Baire category and measurable functions.
• To analyze Lusin’s 1934 Comptes Rendus note, which introduced a deep result on non-separable sets that cannot be separated by Borel sets, and which remained largely unnoticed.
• To show that Kunugi’s approach, based on a theorem of Lusin and Novikov, provides a solution to Kuratowski’s 1935 problem on extending the continuity a.e. property to non-separable codomains.
• To demonstrate that the Baire property of a function into a metric space implies continuity outside a first category set, even when the codomain is not separable.
• To provide a modern exposition and proof of these forgotten results, highlighting their relevance to contemporary research in descriptive set theory and measure theory.

Proposed method

• Reconstructs Kunugi’s 1936 proof using a decomposition of metric spaces into $G_ ho$-sets with positive mutual distance, leveraging a method inspired by Montgomery’s work.
• Applies Lusin’s 1934 result: if a set $F \subset \mathbb{R}$ is not always first category, then it can be split into two disjoint subsets $F_1, F_2$ that cannot be separated by any Borel set.
• Uses the concept of 'envelopes' and outer measure to show that in a perfect set $Q$ of positive measure, two disjoint subsets $E_1, E_2$ can each have outer measure equal to $m(Q)$.
• Applies the notion of '2nd category at a point' and 'nowhere dense' sets to characterize the residual structure of sets in topological spaces.
• Establishes that if a function $f: Y \to X$ has the Baire property and $Y$ is Čech complete of weight $\leq \mathfrak{c}$, then $f$ is continuous outside a first category set.
• Proves a strengthening of the main result: for point-finite families of universally $\mathcal{M}$-null or always first category sets, if the union is not null, then no measurable subset can separate a subfamily from the rest.

Experimental results

Research questions

• RQ1 Can the separability assumption in the classical result that 'a function with the Baire property is continuous outside a first category set' be removed?
• RQ2 What is the significance of Kunugi’s 1936 paper in the context of Kuratowski’s 1935 problem on extending the Baire property result to non-separable codomains?
• RQ3 How does Lusin’s 1934 result on non-separable sets that cannot be separated by Borel sets relate to the structure of measurable and Baire sets?
• RQ4 To what extent do the concepts of 'always first category' and '2nd category at a point' help in analyzing the structure of subsets of topological spaces?
• RQ5 Can the non-separable case of the Baire property continuity theorem be proven using only topological and measure-theoretic tools, without forcing or model-theoretic methods?

Key findings

• Kunugi’s 1936 paper provides a solution to Kuratowski’s 1935 problem, showing that a function with the Baire property from a metric space into a topological space is continuous outside a first category set, even when the codomain is not separable.
• Lusin’s 1934 result establishes that there exist disjoint subsets of a perfect set of positive measure that cannot be separated by any Borel set, a result foundational for understanding non-separable Baire sets.
• The paper proves that if $Y$ is a Čech complete space of weight $\leq \mathfrak{c}$, then any function $f: Y \to X$ with the Baire property is continuous outside a first category set, even if $X$ is not separable.
• The authors show that for a point-finite family of universally $\mathcal{M}$-null or always first category sets, if their union is not null, then no universally $\mathcal{M}$-measurable set can separate a subfamily from the rest.
• In the presence of a measure vanishing on points, a perfect set $Q$ of positive measure can be found such that two disjoint subsets $E_1, E_2$ of $Q$ each have outer measure equal to $m(Q)$, confirming Lusin’s claim.
• The paper establishes that if $A \subset E$ has the Baire property relative to $E$, and $E$ is dense in $X$, then $A$ has the Baire property relative to $E$, under certain conditions, which is used to derive a contradiction in the proof of non-separability.