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Integration by parts for absolutely continuous functions
Statement
Assume the Axioms of Countable Choice and Dependent Choice. For ,
Facts & Assumptions
Given: Countable choice, dependent choice, and absolutely continuous functions on .
Proof
The product lemma makes AC. At the common full-measure set where and exist, the ordinary product rule gives . The sharp FTC makes integrable, and continuity on the compact interval bounds , so both products are integrable.
Apply Fundamental theorem of calculus for absolutely continuous functions to and integrate the displayed derivative.
Rearranging gives the stated identity, with identical zero sides on .
Depends on
- The Axiom of Countable Choice ($\mathrm{AC}_\omega$)
- The axiom of dependent choice: a relation in which every element is related to something admits an $\mathbb{N}$-indexed chain
- The product of two absolutely continuous functions is absolutely continuous
- Fundamental theorem of calculus for absolutely continuous functions
- Fubini's theorem for L^1 functions on a sigma-finite product
Used by
- A symmetric closed operator that is not self-adjoint Counterexample
- A boundary flux control uses the H1 trace and a different control operator Example
- A one-dimensional obstacle problem and its contact set Example
- An integral constraint and its constant multiplier Example
- Integration by parts for absolutely continuous functions Example
- Periodic derivative and its unitary translation group Example
- The one-dimensional obstacle reaction is supported on the contact set Example
- The weak Dirichlet Poisson problem on an interval Example
- ACL representatives recover their weak gradients by Fubini Lemma
- Sharp Poincare bound for the uniform law on an interval Lemma
- Stationary phase with a compactly supported amplitude Lemma
- The LQR adjoint identity gives the reduced gradient Lemma
- Two-sided Mills bounds for the standard normal tail Lemma
- Finite-dimensional Gaussian isoperimetric expansion Theorem
- Gross logarithmic Sobolev inequality for standard Gaussian measure Theorem
Dependency tree · two levels
28 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
- Donald L. Cohn, Measure Theory, 2nd ed., Corollary 6.3.9 (standard reference, not scraped)