Finite, sigma-finite, and semifinite measures

Depends on

• Measure spaces
• Measures on sigma-algebras
• Finite, countably infinite, countable, uncountable

Used by

• Dominated convergence is a Vitali corollary Corollary
• Measurable Hilbert field from a countable fundamental family Definition
• The semifinite part of a measure Definition
• A Hilbert–Schmidt kernel operator is compact on L two Example
• A square-integrable separable product kernel Example
• Adjoints of shifts, multiplication and integral operators Example
• Assuming countable choice, counting measure is sigma-finite exactly on countable sets Example
• Assuming countable choice, zero on countable sets and infinity on cocountable sets is a non-semifinite measure Example
• Finite-rank truncations of a square-integrable kernel Example
• Integral operator trace under a valid diagonal hypothesis Example
• Multiplicity-two diagonal representation Example
• Pvm of a multiplication operator Example
• The square of the Volterra operator has zero trace Example
• FALSE: Tonelli's theorem still holds without any sigma-finiteness hypothesis False statement
• Assuming countable choice, an infinite-measure set in a semifinite measure space has arbitrarily large finite-measure subsets Lemma
• Complex Lq norm recovery from finite simple dual tests Lemma
• Finite-measure sets are approximable in measure by sets from a countable generating algebra Lemma
• For 1 ≤ p < ∞, every Lᵖ(μ) class has a sigma-finite essential support Lemma
• Product rectangle kernels are dense in product L two Lemma
• Sigma-finite ergodic oscillation sets have finite measure Lemma
• For sigma-finite measures, the section-measure functions are measurable Proposition
• Lebesgue measure is sigma-finite, and every metrically bounded subset of ℝⁿ has finite outer measure Proposition
• The functional Λ_g has norm ‖g‖_q; for q=∞ assume μ is semifinite Proposition
• Assuming countable choice, every measure is the sum of its semifinite part and a zero-infinity-valued measure Theorem
• Chain rule for a C¹ function with bounded derivative Theorem
• Chain rule for globally Lipschitz scalar maps of Sobolev functions Theorem
• Decomposable operators are the commutant of diagonal multiplication Theorem
• Direct integrals of measurable Hilbert fields are Hilbert spaces Theorem
• For 1 < p < ∞, the same representation theorem holds on arbitrary measure spaces Theorem
• For sigma-finite factors, the product measure exists, has the rectangle formula, is sigma-finite, and is unique Theorem
• For sigma-finite measures, the two section-measure integrals of a measurable set agree Theorem
• If μ is sigma-finite and A is countably generated, then Lᵖ(μ) is separable for 1 ≤ p < ∞ Theorem
• L two kernels give Hilbert–Schmidt operators Theorem
• Measurable essentially bounded operator fields act decomposably Theorem
• Measures agreeing on a generating pi-system are equal under an increasing finite-measure exhaustion from that pi-system Theorem
• On a sigma-finite measure space, every bounded linear functional on Lᵖ is integration against a unique L^q function Theorem
• Spectral multiplicity model for separably acting abelian von Neumann algebras Theorem
• Vitali convergence theorem on finite and sigma-finite measure spaces Theorem

Dependency tree · two levels

11 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

• G. Folland, Real Analysis, 2nd ed., §1.3 (standard reference, not scraped)