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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- AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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These labels describe origin, not correctness: citations and verification chips remain separate evidence.
A positive, signed, or complex measure concentrated on a measurable set
Definition
Let be a positive measure, a signed measure, or a complex measure on , and let . We say that is concentrated on if Equivalently, every measurable subset of has -value . For a positive measure this is the same as , but for a signed or complex measure the stronger subsetwise vanishing is the load-bearing form used in later proofs.
Depends on
Used by
- 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
- The Lebesgue decomposition of a sigma-finite signed measure is unique 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, Chapter 13 (standard reference, not scraped)
- John K. Hunter, Measure Theory, §6.8 (standard reference, not scraped)