The modulus of an integral is bounded by the integral of the modulus

Depends on

• Integrable real and complex functions, and their integrals
• The Lebesgue integral is linear on $L^1(\mu)$
• Real and imaginary parts, complex conjugation, and modulus
• Monotonicity and nonnegative homogeneity of the nonnegative integral

Used by

• Linearity, monotonicity, and the modulus bound for expectation Corollary
• One-dimensional W^1,p functions have unique absolutely continuous representatives Corollary
• A step has no locally integrable weak derivative Counterexample
• Cantor function has singular distributional derivative Counterexample
• Strong fractional integration fails at p equal to one Counterexample
• Characteristic function of a real random variable Definition
• The Poisson integral of a finite complex boundary measure Definition
• Dyadic conditional expectation martingale Example
• Interpolation of an averaging operator on a probability space Example
• Newton shell theorem from harmonic mean values Example
• Principal value one over x is tempered and its fourier transform Example
• Uniform laws on expanding finite grids converge to uniform zero one Example
• A Reiter net has an invariant-mean cluster point Lemma
• A topological invariant mean yields norm-approximately invariant densities Lemma
• A UCB-invariant mean yields a topological invariant mean Lemma
• Basic properties of characteristic functions Lemma
• Compact support gives an entire Fourier-Laplace transform by slices Lemma
• Complex l one functionals on finite measure spaces have bounded densities Lemma
• Complex Lq norm recovery from finite simple dual tests Lemma
• Complex translation, convolution, approximate identities, and mollification Lemma
• Distribution pairing with smooth parameter families Lemma
• Fourier-Stieltjes transforms of positive measures are continuous positive definite Lemma
• Gaussian decay gives an entire Fourier-Laplace transform and its growth bound Lemma
• Hedberg pointwise inequality for Riesz potentials Lemma
• L1 convolution smooths bounded functions into UCB Lemma
• Probability-density approximation of continuous tests and topological means Lemma
• Probability-density averages and locally detectable upper essential values Lemma
• Reiter functions can be cut down to Følner sets Lemma
• Schwartz parameter pairing and integral interchange Lemma
• Sigma-finite ergodic oscillation sets have finite measure Lemma
• Submultiplicativity of convolution in the L1 norm Lemma
• The integral transform is representative independent Lemma
• The maximal ergodic inequality on a probability space Lemma
• Compact and locally compact abelian groups are amenable Proposition
• A complex L¹ density defines a complex measure whose total variation is |h| dmu Theorem
• Birkhoff's theorem for an ergodic probability system Theorem
• Complex Holder, Minkowski, and the quotient norm Theorem
• Dominated convergence Theorem
• Far-field asymptotics of compact-source Newtonian potentials Theorem
• Feller converse to Lindeberg-Feller Theorem

…and 14 more results.

Dependency tree · two levels

12 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

• Gerald B. Folland, Real Analysis, 2nd ed., Proposition 2.22 (standard reference, not scraped)
• John K. Hunter, Measure Theory Notes, Proposition 4.9 (standard reference, not scraped)