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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These labels describe origin, not correctness: citations and verification chips remain separate evidence.
Absolute continuity of a signed or complex measure with respect to a positive measure
Definition
Let be a positive measure and let be a signed measure or a complex measure on the same measurable space . We say that is absolutely continuous with respect to , and write if every measurable -null set is also -null: Because every measurable subset of a -null set is again -null, this is equivalent to requiring for every measurable whenever .
Depends on
Used by
- FALSE: the epsilon-delta condition characterises absolute continuity for every measure False statement
- A signed or complex measure that is both absolutely continuous and singular with respect to the same positive measure is zero Theorem
- Every sigma-finite signed measure admits a Lebesgue decomposition relative to a sigma-finite positive measure Theorem
- For signed and complex measures, absolute continuity is equivalent for the measure, its Jordan or real-imaginary parts, and its total variation Theorem
Dependency tree · two levels
8 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
- Richard F. Bass, Real Analysis for Graduate Students, Definition 13.1 (standard reference, not scraped)
- John K. Hunter, Measure Theory, Definition 6.22 (standard reference, not scraped)
- Sheldon Axler, Measure, Integration & Real Analysis, 9.32 (standard reference, not scraped)