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Absolute continuity of the integral
Statement
Let and let . Then there is such that for every measurable ,
Facts & Assumptions
Given: An integrable function and a real number .
The truncations increase pointwise to , so their integrals converge to by monotone convergence (Monotone convergence for the integral).
The integral over a measurable set is defined by (Integral over a measurable subset).
The nonnegative integral is monotone and homogeneous (Monotonicity and nonnegative homogeneity of the nonnegative integral).
Integrability means (Integrable real and complex functions, and their integrals).
The nonnegative integral is additive (Additivity of the nonnegative Lebesgue integral).
Proof
Put . The pointwise identity and [L5] give ; the subtraction is valid because both integrals are finite by [L4]. By [L1] choose so large that , and put .
If , then [L2], [L3], and [L5] give . The last inequality follows from .
Depends on
Used by
- One-dimensional W^1,p functions have unique absolutely continuous representatives Corollary
- The indefinite integral of an L¹ function is absolutely continuous Corollary
- Weak differentiation has a closed graph on its natural domains Corollary
- A hypersurface jump is not W^1,p Counterexample
- A step has no locally integrable weak derivative Counterexample
- Bounded discontinuous elliptic coefficients need not give H² solutions Counterexample
- Dunford--Pettis: dominated and concentrating families Example
- The weak Dirichlet Poisson problem on an interval Example
- Continuous functions are dense in Lᵖ of finite tori and of bounded intervals Lemma
- L¹ of an LCA group is a commutative Banach star algebra under convolution Lemma
- Scalar unitisation of L¹ of an LCA group: characters, spectrum and identity criterion Lemma
- The variation-of-constants integral is continuous for integrable forcing Lemma
- Dominated families are uniformly integrable Proposition
- At p=1 bounded difference quotients need not give an L¹ weak derivative Remark
- Weak derivatives are represented distributional derivatives Remark
- A right-continuous nondecreasing function splits uniquely as absolutely continuous plus jump plus singular continuous Theorem
- Banach--Zarecki characterisation of absolute continuity Theorem
- Dunford--Pettis for real L¹ on a finite measure space Theorem
- For finite signed or complex measures, absolute continuity is equivalent to the epsilon-delta small-set condition Theorem
- Uniform integrability of conditional expectations of one variable Theorem
- Vitali convergence theorem on finite and sigma-finite measure spaces Theorem
Dependency tree · two levels
13 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 4.16 (standard reference, not scraped)