[Paper Review] A normal measure on a compact connected space

Grzegorz Plebanek|arXiv (Cornell University)|Jul 10, 2015

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TL;DR

This paper constructs a compact connected space of weight 𝔠 that supports a normal probability measure, resolving a long-standing question by demonstrating that normal measures can exist on connected compacta. The construction uses an inverse limit of iterated extensions via a strictly positive measure-preserving surjection, ensuring every positive-measure zero set has non-empty interior, thus satisfying normality via a criterion from measure theory.

ABSTRACT

We present a construction of a compact connected space which supports a normal probability measure.

Motivation & Objective

• To resolve the open problem of whether a compact connected space can support a normal probability measure.
• To construct a compact, connected space of topological weight 𝔠 that supports a normal measure.
• To demonstrate that normality of a measure does not imply disconnectedness of the underlying space.
• To extend previous results showing that locally connected compacta cannot support normal measures.

Proposed method

• An inverse system of compact connected spaces is constructed using transfinite recursion of length ω₁, starting from [0,1] with Lebesgue measure.
• At each successor stage α+1, the space L_{α+1} is defined as the 'extension' (L_α)^# via Lemma 3.2, ensuring every zero set of positive measure in L_α has a preimage with non-empty interior in L_{α+1}.
• The limit space L is taken as the inverse limit of the system, with the limit measure ν defined via the universal property of inverse limits.
• The construction ensures that every closed G_δ set Z ⊆ L with ν(Z) > 0 has non-empty interior, satisfying the criterion for normality via Lemma 2.1.
• Topological weight is preserved at each stage, ensuring w(L) = 𝔠.
• The key technical tool is Lemma 3.2, which ensures that preimages of positive-measure zero sets have non-empty interior in the extended space.

Experimental results

Research questions

• RQ1 Can a compact connected space support a normal probability measure, despite the fact that locally connected compacta cannot?
• RQ2 Is there a compact connected space of weight 𝔠 that supports a normal measure?
• RQ3 Does the existence of a normal measure on a compact space imply that the space must be disconnected?
• RQ4 Can the construction be refined to yield a perfectly normal space under the continuum hypothesis?
• RQ5 What topological and measure-theoretic conditions ensure that a measure is normal on a compact space?

Key findings

• A compact connected space L of weight 𝔠 exists that supports a normal probability measure, thus answering Problem 1.1 in the negative.
• The measure ν on L is normal because every closed G_δ set Z ⊆ L with ν(Z) > 0 has non-empty interior, satisfying the criterion from Lemma 2.1.
• The construction ensures that ν is strictly positive and order-continuous on C(L), confirming its normality.
• The space L is the inverse limit of a system of length ω₁, with each stage preserving connectedness and measure-theoretic properties.
• The topological weight of L is exactly 𝔠, as the construction preserves weight at each stage.
• Under the continuum hypothesis, a variant construction yields a perfectly normal, first-countable compact connected space supporting a normal measure (Theorem 3.3).