The nonnegative integral agrees with the simple integral on simple functions

Depends on

• The nonnegative Lebesgue integral
• The integral of a nonnegative simple function
• The simple integral is independent of the chosen representation
• The simple integral is monotone, homogeneous, and additive

Used by

• A nonnegative measurable function with finite integral is finite almost everywhere Corollary
• Additivity of the nonnegative Lebesgue integral Corollary
• Polar integration may discard the cut locus Corollary
• The expectation of an indicator is the probability of the event Corollary
• Weak differentiation has a closed graph on its natural domains Corollary
• A nonintegrable observable with divergent ergodic averages Counterexample
• A random variable need not have a finite expectation Counterexample
• A step has no locally integrable weak derivative Counterexample
• Counting measure on [0,1] shows the dominating measure needs sigma-finiteness Counterexample
• Neumann Poisson data require a flux compatibility equation Counterexample
• Point evaluation is unbounded below the Sobolev continuity threshold Counterexample
• Sharp frequency cutoffs have kernels that are not in L1 Counterexample
• Strong fractional integration fails at p equal to one Counterexample
• The critical Riesz potential can diverge and be essentially unbounded Counterexample
• Rademacher functions on the unit interval Definition
• Compact groups have a constant Reiter net Example
• Counting measure represents finite-support summation on a discrete LCH space Example
• Hilbert transform of the line Poisson kernel Example
• Integrating against counting measure recovers a series Example
• Matching C¹ pieces across a hyperplane have no jump derivative Example
• Mollification of a complex two-step function Example
• Negative drift gives a finite mean small-set hit Example
• Sharp Sobolev threshold for a radial power Example
• The absolute value has a weak first derivative Example
• The Haar orthonormal basis of L²((0,1)) Example
• Uniform laws on expanding finite grids converge to uniform zero one Example
• Birkhoff's theorem requires integrability False statement
• A bounded Riemann integrable function admits Borel Darboux envelopes with the same Lebesgue integral Lemma
• A topological invariant mean yields norm-approximately invariant densities Lemma
• Averaging a Hermitian form unitarizes a finite-dimensional compact-group representation Lemma
• Borel Darboux integrands in finite dimension Lemma
• Counting measure on a discrete group is Haar, Haar measures there are its multiples, and integrals against them are sums Lemma
• Finite Haar mass, compact detection, and integrable pairings Lemma
• Finite Rademacher blocks are equidistributed Lemma
• Følner nets give Reiter nets Lemma
• Probability-density averages and locally detectable upper essential values Lemma
• Riemann–Lebesgue comparison for distribution test integrands Lemma
• Two common diagonalizations differ by a bimeasurable base isomorphism and a measurable field of unitaries Lemma
• Compact and locally compact abelian groups are amenable Proposition
• Haar integration is translation and conjugation invariant Proposition

…and 21 more results.

Dependency tree · two levels

7 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, Definition 4.4 (standard reference, not scraped)