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The measure with density relative to
Definition
Let be a measure space and let be measurable. The measure with density relative to is the measure whose measure property is supplied by The indefinite integral of a nonnegative measurable function is a measure.
Depends on
Used by
- A homogeneous quotient without invariant measure Counterexample
- An absolutely continuous finite measure can have an unbounded Radon-Nikodym derivative Counterexample
- Counting measure on [0,1] shows the dominating measure needs sigma-finiteness Counterexample
- x⁻¹dλ on (0,1) shows finiteness is needed in the epsilon-delta criterion Counterexample
- A locally integrable density functional is represented by g dlambda Example
- A piecewise-quadratic distribution function recovers its density Example
- The density 2x on [0,1] is the Radon-Nikodym derivative of its density measure Example
- FALSE: absolutely continuous measures always have bounded Radon-Nikodym derivatives False statement
- FALSE: the epsilon-delta condition characterises absolute continuity for every measure False statement
- FALSE: the Radon-Nikodym theorem holds without sigma-finiteness False statement
- Haar change of variables under inversion Lemma
- Local polynomial projections matching moments through order s Lemma
- The weighted discrete-series space is a Hilbert space with K-type basis Lemma
- Every sigma-finite signed measure admits a Lebesgue decomposition relative to a sigma-finite positive measure Theorem
- Integrating against a density agrees with integrating the product Theorem
- Poisson extension is an Lp contraction and converges in finite Lp 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
- Gerald B. Folland, Real Analysis, 2nd ed., §2.2 (standard reference, not scraped)