Alphabeta MathHow statement and proof provenance work
The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.
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The Cantor measure is atomless but not absolutely continuous with respect to Lebesgue measure
Statement refuted
That every atomless Borel measure on must be absolutely continuous with respect to Lebesgue measure.
Facts & Assumptions
Given: The Axiom of Countable Choice and the Cantor measure .
Assuming Countable Choice, the Cantor measure is an atomless probability measure singular with respect to Lebesgue measure. (The Cantor measure is a singular atomless probability measure concentrated on the Cantor set)
Counterexample
By [L1], the measure is atomless.
The same fact [L1] says that is singular and has total mass . [step 1.1, L1] If it were also absolutely continuous with respect to Lebesgue measure, its concentration on a Lebesgue-null set would force its total mass to be , a contradiction. Therefore atomlessness does not imply absolute continuity.
Depends on
Used by
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Dependency tree · two levels
7 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- John K. Hunter, Measure Theory, Example 2.37 (standard reference, not scraped)