Two integrable functions are equal almost everywhere exactly when all of their indefinite integrals agree

Depends on

• A nonnegative integral over a null set vanishes
• The Lebesgue integral is linear on $L^1(\mu)$
• A nonnegative measurable function has integral $0$ exactly when it vanishes almost everywhere
• Integrable real and complex functions, and their integrals

Used by

• Dominated convergence is a Vitali corollary Corollary
• One-dimensional W^1,p functions have unique absolutely continuous representatives Corollary
• Polar integration may discard the cut locus Corollary
• Pointwise shock values do not affect the weak solution Counterexample
• BMO seminorm and the quotient by constants Definition
• Conditional expectation as an ae class Definition
• Direct integral of a measurable Hilbert field Definition
• Fourier coefficients and trigonometric polynomials on the torus Definition
• Natural and usual augmented Brownian filtrations Definition
• The heat evolution Hₜ of initial data Definition
• The Lusin area function for a fixed admissible kernel and aperture Definition
• The Poisson integral of a finite complex boundary measure Definition
• Compact groups have a constant Reiter net Example
• Piecewise-affine approximation of a measurable coefficient Example
• Projective-plane curvature via a hemisphere Example
• FALSE: every increasing function satisfies Newton-Leibniz with its derivative False statement
• Centring by translation and modulation preserves the variance product Lemma
• Compact support gives an entire Fourier-Laplace transform by slices Lemma
• Complex translation, convolution, approximate identities, and mollification Lemma
• Conditional variance is well-defined and has the second-moment formula Lemma
• Convolution on L¹(ℝⁿ) is independent of the chosen Borel representatives Lemma
• Expectation depends only on the almost-everywhere class Lemma
• Finite Rademacher blocks are equidistributed Lemma
• Gaussian decay gives an entire Fourier-Laplace transform and its growth bound Lemma
• L² with the integral pairing is a Hilbert space Lemma
• Near and far bounds for a Riesz potential Lemma
• Real L2 multipliers and unitary transport Lemma
• Sobolev functions paste across an overlap Lemma
• The integral transform is representative independent Lemma
• Weak differentiation ignores null-set changes Lemma
• Compact and locally compact abelian groups are amenable Proposition
• Dominated families are uniformly integrable Proposition
• A bounded Riemann integrable function on a closed bounded interval is Lebesgue measurable and has the same integral Theorem
• Complex Holder, Minkowski, and the quotient norm Theorem
• Every g∈ L^q(μ) defines a bounded linear functional on Lᵖ(μ) Theorem
• For a nondecreasing function, the derivative is measurable and integrable and its integral is bounded by the total increase Theorem
• Gibbs overshoot at a piecewise C¹ jump Theorem
• Hardy–Littlewood–Sobolev fractional integration inequality Theorem
• Hᵏ is a Hilbert space under the derivative-sum inner product Theorem
• Kolmogorov convergence criterion Theorem

…and 5 more results.

Dependency tree · two levels

14 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, Proposition 8.2 (standard reference, not scraped)
• Gerald B. Folland, Real Analysis, 2nd ed., Proposition 2.23(b) (standard reference, not scraped)