Additivity of the nonnegative Lebesgue integral

Depends on

• Monotone convergence for the integral
• Every nonnegative measurable function is the increasing limit of simple measurable functions
• Closure properties of measurable functions used by the integral
• The simple integral is monotone, homogeneous, and additive
• The nonnegative integral agrees with the simple integral on simple functions

Used by

• A C¹ diffeomorphism satisfies the change-of-variables formula for L¹ functions Corollary
• Almost-everywhere monotone convergence Corollary
• Beppo Levi's theorem for nonnegative series Corollary
• Polar integration may discard the cut locus Corollary
• Reverse Fatou's lemma under an integrable majorant Corollary
• Strong fractional integration fails at p equal to one Counterexample
• The critical Riesz potential can diverge and be essentially unbounded Counterexample
• Direct integral of a measurable Hilbert field Definition
• The standard intertwining operator A(nu) Definition
• Direct integral of a constant Hilbert field Example
• Poisson extension of an indicator arc Example
• The exponential tail function is integrable by monotone truncation and geometric comparison Example
• The function x^-1/2 on (0,1] is unbounded and integrable Example
• The positive-type Gaussian on the real line and its cyclic model Example
• The square of the Volterra operator has zero trace Example
• Bochner integral norm inequality Lemma
• Clarkson inequalities in both exponent ranges Lemma
• Compactness, finite Haar volume and invariant vectors in the regular representation Lemma
• Completeness of the complex Haar L1 and L2 spaces and density of Cc Lemma
• Hedberg pointwise inequality for Riesz potentials Lemma
• Near and far bounds for a Riesz potential Lemma
• Sobolev functions paste across an overlap Lemma
• The integral of the divergence of an integrable C1 field vanishes Lemma
• The level-set kernel measure estimate for the Slobodeckij kernel Lemma
• The rho-length and the extremal length are well defined Lemma
• A C¹ diffeomorphism satisfies the change-of-variables formula for nonnegative Lebesgue measurable functions Theorem
• A real L¹ density defines a finite signed measure with its canonical Hahn and Jordan data Theorem
• Absolute continuity of the integral Theorem
• Bochner integrability criterion Theorem
• Chebyshev-Markov inequality for the integral Theorem
• Decomposable operators are the commutant of diagonal multiplication Theorem
• Dominated convergence Theorem
• Equality in Minkowski's inequality for 1 < p < ∞ Theorem
• Every sigma-finite signed measure admits a Lebesgue decomposition relative to a sigma-finite positive measure Theorem
• First-step equations for nonnegative exit costs Theorem
• For sigma-finite factors, the product measure exists, has the rectangle formula, is sigma-finite, and is unique Theorem
• Hardy–Littlewood–Sobolev fractional integration inequality Theorem
• Holder's inequality for integrals, including the endpoint cases Theorem
• Integrable simple functions are dense in L¹(μ) Theorem
• Integrating against a density agrees with integrating the product Theorem

…and 12 more results.

Dependency tree · two levels

17 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 Notes, Proposition 4.7 (standard reference, not scraped)
• Gerald B. Folland, Real Analysis, 2nd ed., Theorem 2.15 (standard reference, not scraped)