Fundamental theorem of calculus for absolutely continuous functions

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 indefinite integral of an $L^1$ function is absolutely continuous
• The indefinite integral of an $L^1$ function is differentiable almost everywhere
• Absolutely continuous functions have integrable derivatives
• An absolutely continuous function with zero derivative almost everywhere is constant

Used by

• One-dimensional W^1,p functions have unique absolutely continuous representatives Corollary
• A symmetric closed operator that is not self-adjoint Counterexample
• The composition of two absolutely continuous functions need not be absolutely continuous Counterexample
• x² sin(1/x²) is differentiable everywhere but not absolutely continuous Counterexample
• Convex entropy--entropy flux pairs Definition
• The ACL and Sobolev analytic definition of quasiconformality Definition
• A one-dimensional obstacle problem and its contact set Example
• An integral constraint and its constant multiplier Example
• Periodic derivative and its unitary translation group Example
• The Neumann Laplacian has a zero constant mode Example
• The weak Dirichlet Poisson problem on an interval Example
• ACL representatives recover their weak gradients by Fubini Lemma
• Analytic quasiconformality gives both quadrilateral modulus bounds Lemma
• Brownian step-potential resolvent at zero Lemma
• Caputo and Volterra formulations are equivalent for absolutely continuous solutions 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
• Exterior finite-energy functions have a trace controlled by their gradient Lemma
• Green identity and Kirchhoff flux cancellation on a metric graph Lemma
• Integrated pressure recovery and MINI pressure-primitive convergence Lemma
• Sobolev pullback and Green identity on Lipschitz graphs Lemma
• Solutions of projected contact dynamics are unique under the inherited choice assumptions Lemma
• The inverse of a quasiconformal map is quasiconformal with the same dilatation Lemma
• The translation estimate for W^1,p functions on ℝⁿ Lemma
• Whole-space support and the zero-boundary Sobolev closure Lemma
• Sharp and classical fundamental theorems of calculus agree Remark
• A posteriori ODE error bound from the continuous residual Theorem
• A square-root global error bound for projected Euler with certified projection tolerances Theorem
• Alexandrov second differentiability of locally semiconvex functions Theorem
• Chain rule for globally Lipschitz scalar maps of Sobolev functions Theorem
• Change of variables for an absolutely continuous map under an absolutely continuous composition hypothesis Theorem
• Change of variables for an increasing absolutely continuous function Theorem
• Compactness of the normalized K-quasiconformal self-maps of the sphere Theorem
• Countably exceptional differentiability and integrable derivative imply absolute continuity Theorem
• Existence and uniqueness for projected dynamics and convergence of projection Euler under countable and dependent choice Theorem
• Finite-dimensional Gaussian isoperimetric expansion Theorem
• Gross logarithmic Sobolev inequality for standard Gaussian measure Theorem
• Higher-order Rellich--Kondrachov compactness Theorem
• Integration by parts for absolutely continuous functions Theorem
• Lipschitz characterisation within absolutely continuous functions Theorem

…and 11 more results.

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

• Christopher Heil, Absolute Continuity and the Banach--Zaretsky Theorem, Theorem 23 (standard reference, not scraped)