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The indefinite integral of an function is absolutely continuous
Statement
For , the function is absolutely continuous.
Facts & Assumptions
Given: and its indefinite integral .
Proof
Given , apply Absolute continuity of the integral to and obtain .
For disjoint intervals of total length below , their union has measure below , and .
This is the defining finite-family condition for absolute continuity; the empty family and give zero sums.
Depends on
Used by
- One-dimensional W^1,p functions have unique absolutely continuous representatives Corollary
- A symmetric closed operator that is not self-adjoint Counterexample
- Change of variables through an increasing AC map with a positive-measure flat set Example
- Integration by parts for absolutely continuous functions Example
- Periodic derivative and its unitary translation group Example
- The weak Dirichlet Poisson problem on an interval Example
- AC zero-derivative constancy and integrable-derivative reconstruction under Countable Choice Lemma
- Brownian step-potential resolvent at zero Lemma
- Chain rule for an indefinite integral after an absolutely continuous composition Lemma
- Compact subsets of lines and round circles are removable for quasiconformal maps Lemma
- Inter-event survival and density from a conditional intensity Lemma
- Positive L1 weights, discrete comparison, and residual stability Lemma
- The bounded measurable linear ODE has a unique affine state map Lemma
- A posteriori ODE error bound from the continuous residual Theorem
- Fundamental theorem of calculus for absolutely continuous functions Theorem
- Integrated Legendre projection gives an explicit p-rate in H1 Theorem
- The ACL characterisation of W^1,p Theorem
- The continuous survival and cumulative-hazard identity Theorem
- The Hamilton--Jacobi correspondence in one dimension Theorem
- Total-variation function of an absolutely continuous function Theorem
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
- Christopher Heil, Absolute Continuity and the Banach--Zaretsky Theorem, Lemmas 14 and 22 (standard reference, not scraped)