[Paper Review] Good measures on locally compact Cantor sets

Olena Karpel|arXiv (Cornell University)|Mar 30, 2012

Advanced Topology and Set Theory13 references3 citations

TL;DR

This paper classifies full, non-atomic Borel measures on non-compact, locally compact Cantor sets up to homeomorphism by introducing the concepts of 'good' measures and the compact open values set $ S(\mu) $. It establishes a complete invariant for homeomorphism classes of good measures and constructs examples with prescribed values sets and defective sets, extending results from compact Cantor sets to the non-compact case.

ABSTRACT

We study the set M(X) of full non-atomic Borel (finite or infinite) measures on a non-compact locally compact Cantor set X. For an infinite measure $μ$ in M(X), the set $\mathfrak{M}_μ= \{x \in X : {for any compact open set} U i x {we have} μ(U) = \infty \}$ is called defective. We call $μ$ non-defective if $μ(\mathfrak{M}_μ) = 0$. The class $M^0(X) \subset M(X)$ consists of probability measures and infinite non-defective measures. We classify measures $μ$ from $M^0(X)$ with respect to a homeomorphism. The notions of goodness and compact open values set $S(μ)$ are defined. A criterion when two good measures from $M^0(X)$ are homeomorphic is given. For any group-like $D \subset [0,1)$ we find a good probability measure $μ$ on X such that $S(μ) = D$. For any group-like $D \subset [0,\infty)$ and any locally compact, zero-dimensional, metric space A we find a good non-defective measure $μ$ on X such that $S(μ) = D$ and $\mathfrak{M}_μ$ is homeomorphic to A. We consider compactifications cX of X and give a criterion when a good measure $μ\in M^0(X)$ can be extended to a good measure on cX.

Motivation & Objective

• To extend the classification of good measures from compact to non-compact, locally compact Cantor sets.
• To define and analyze non-defective infinite measures, introducing the defective set $ \mathfrak{M}_{\mu} $ as a key invariant.
• To establish a complete invariant for homeomorphism classes of good measures on non-compact locally compact Cantor sets.
• To construct measures with prescribed compact open values sets $ S(\mu) $, including for arbitrary group-like subsets of $[0,1)$ and $[0,\infty)$.
• To investigate the behavior of good measures under compactification, particularly when extending to Cantor space compactifications.

Proposed method

• Define a measure $ \mu \in M(X) $ as non-defective if $ \mu(\mathfrak{M}_{\mu}) = 0 $, where $ \mathfrak{M}_{\mu} $ is the set of points whose every compact open neighborhood has infinite $ \mu $-measure.
• Introduce the compact open values set $ S(\mu) $ as the set of all finite $ \mu $-measures of compact open subsets of $ X $.
• Define a measure as 'good' if, for any clopen sets $ U, V $ with $ \mu(U) < \mu(V) $, there exists a clopen $ W \subset V $ such that $ \mu(W) = \mu(U) $.
• Use the structure of a countable basis of compact open sets to construct measures with specified $ S(\mu) $, leveraging group-like subsets of $[0,1)$ and $[0,\infty)$.
• Construct examples via $ (C,F) $-constructions and $ p $-adic measures to realize arbitrary good $ S(\mu) $ and $ \mathfrak{M}_{\mu} $ homeomorphic to any given zero-dimensional, locally compact, metric space $ A $.
• Analyze compactifications $ cX $ of $ X $, restricting to those where $ cX $ is a Cantor set, and study when a good measure on $ X $ extends to a good measure on $ cX $.

Experimental results

Research questions

• RQ1 When are two good measures on a non-compact locally compact Cantor set homeomorphic?
• RQ2 What are the necessary and sufficient conditions for a good measure to extend to a good measure on a compactification of the space?
• RQ3 For a given group-like subset $ D \subset [0,1) $, does there exist a good probability measure $ \mu $ with $ S(\mu) = D $?
• RQ4 For a given group-like subset $ D \subset [0,\infty) $ and any zero-dimensional, locally compact, metric space $ A $, does there exist a good non-defective measure $ \mu $ with $ S(\mu) = D $ and $ \mathfrak{M}_{\mu} \cong A $?
• RQ5 How does the compact open values set $ S(\mu) $ change under compactification, and when is the good property preserved?

Key findings

• Two good measures on a non-compact locally compact Cantor set are homeomorphic if and only if their compact open values sets $ S(\mu) $ are equal.
• For any group-like subset $ D \subset [0,1) $, there exists a good probability measure $ \mu $ on $ X $ such that $ S(\mu) = D $.
• For any group-like subset $ D \subset [0,\infty) $ and any locally compact, zero-dimensional, metric space $ A $, there exists a good non-defective measure $ \mu $ on $ X $ such that $ S(\mu) = D $ and $ \mathfrak{M}_{\mu} $ is homeomorphic to $ A $.
• The Haar measure on the $ p $-adic numbers $ \mathbb{Q}_p $ is a good measure with $ S(\mu) = \{ np^\gamma \mid n \in \mathbb{N}, \gamma \in \mathbb{Z} \} $.
• The invariant measure for the $ (C,F) $-construction on a non-compact locally compact Cantor set is good, with $ S(\mu) = \{ \frac{a}{|C_1|\cdots|C_n|} \mid a,n \in \mathbb{N} \} \cap [0, \mu(X)) $.
• There exist good ergodic invariant measures on the generating open dense subset of a path space of a stationary Bratteli diagram such that every compactification results in a non-good measure.