Integrable real and complex functions, and their integrals

Depends on

• Closure properties of measurable functions used by the integral
• The nonnegative Lebesgue integral
• Real and imaginary parts, complex conjugation, and modulus

Used by

• A C¹ diffeomorphism satisfies the change-of-variables formula for L¹ functions Corollary
• A finite complex measure absolutely continuous with respect to a sigma-finite positive measure has an integrable complex density Corollary
• Polar integration may discard the cut locus Corollary
• A nonintegrable observable with divergent ergodic averages Counterexample
• A pointwise limit of integrable functions need not be integrable Counterexample
• A step has no locally integrable weak derivative Counterexample
• Infinite variance can defeat square-root-n CLT scaling Counterexample
• The fundamental Hessian is not absolutely locally integrable Counterexample
• The sine integral is improperly Riemann integrable and not Lebesgue integrable Counterexample
• A locally integrable function on ℝⁿ Definition
• A tight family of integrable functions Definition
• A uniformly integrable family Definition
• Characteristic function of a real random variable Definition
• Complex Haar Lᵖ spaces and compactly supported functions Definition
• Complex Lp classes and Euclidean test-function conventions Definition
• Counting, chordal proximity and characteristic Definition
• Direct integral of a measurable Hilbert field Definition
• Expectation of a nonnegative or integrable random variable Definition
• Indefinite Lebesgue integral on a compact interval Definition
• Rademacher functions on the unit interval Definition
• Riesz potential of order alpha Definition
• Spatial and frequency centres and variances of an L² function with finite second moments Definition
• The class L¹(μ) of integrable functions Definition
• The Fourier transform on an LCA group Definition
• The one-dimensional torus and its normalized Haar integral Definition
• The standard intertwining operator A(nu) Definition
• Weak convergence of borel probability measures Definition
• A circle representation with an averaged orthogonal weight form Example
• CLT for sums of uniform random variables Example
• Compact groups have a constant Reiter net Example
• Conjugate phases norm a three-atom function Example
• Integrating against a Dirac measure is evaluation at the point Example
• Matching C¹ pieces across a hyperplane have no jump derivative Example
• Negative drift gives a finite mean small-set hit Example
• Newton shell theorem from harmonic mean values Example
• Newtonian potential of radial compact data Example
• One-dimensional Dirichlet Green kernel on an interval Example
• Point evaluation is represented by a Dirac measure Example
• The positive-type Gaussian on the real line and its cyclic model Example
• The standard inner products make K n, ell two and quotient L two Hilbert spaces Example

…and 78 more results.

Dependency tree · two levels

12 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

• Richard F. Bass, Real Analysis for Graduate Students, Definition 6.2 (standard reference, not scraped)
• John K. Hunter, Measure Theory Notes, Definition 4.8 (standard reference, not scraped)