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.
Mutual singularity for signed or complex measures
Definition
Let and be signed measures or complex measures on the same measurable space . They are mutually singular, written when there are measurable sets with such that:
- every measurable subset of has -value ;
- every measurable subset of has -value .
For positive measures this is equivalent to the usual condition and , because positivity turns vanishing on the ambient set into vanishing on all measurable subsets.
Depends on
Used by
- Pure point, absolutely continuous and singular continuous spectral subspaces Definition
- A signed or complex measure that is both absolutely continuous and singular with respect to the same positive measure is zero Theorem
- Canonical decomposition into pure point, absolutely continuous and singular continuous parts Theorem
- Jordan decomposition of a signed measure into unique mutually singular positive parts Theorem
- The Lebesgue decomposition of a sigma-finite signed measure is unique Theorem
Dependency tree · two levels
6 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, Definition 6.19 (standard reference, not scraped)
- Richard F. Bass, Real Analysis for Graduate Students, Chapter 12 (standard reference, not scraped)