Chebyshev-Markov inequality for the integral

Depends on

• Integral over a measurable subset
• Additivity of the nonnegative Lebesgue integral
• Monotonicity and nonnegative homogeneity of the nonnegative integral
• The nonnegative integral agrees with the simple integral on simple functions
• The integral of a nonnegative simple function

Used by

• Almost-everywhere convergence from the Carleson–Hunt estimate Corollary
• Almost-everywhere convergence of principal-value truncations Corollary
• Convergence in Lᵖ implies convergence in measure Corollary
• Dominated convergence is a Vitali corollary Corollary
• Markov's inequality for random variables Corollary
• The zero-countable / infinity-cocountable measure space breaks the p=1 endpoint of duality Counterexample
• Weak subsolutions and supersolutions of a divergence-form equation Definition
• A function whose trace is at most a level has positive part in the zero-boundary space Lemma
• A weak maximal bound implies almost-everywhere Fourier convergence Lemma
• Calderon reproducing pair and the telescoping identity in S' Lemma
• De Giorgi oscillation reduction: one half-level set is small Lemma
• Differentiation of L-one functions for a doubling weight Lemma
• Ergodic averages converge in Lp on finite-measure spaces Lemma
• For 1 ≤ p < ∞, every Lᵖ(μ) class has a sigma-finite essential support Lemma
• Marcinkiewicz interpolation from weak (1,1) and strong (2,2) Lemma
• Reiter functions can be cut down to Følner sets Lemma
• Sobolev level-set step: energy decay with explicit level gap and radius loss Lemma
• The good part has controlled L2 image Lemma
• The layer-cake identity for integrable functions Lemma
• Dominated families are uniformly integrable Proposition
• The L1 endpoint is excluded Remark
• Calderón–Zygmund operators are bounded on Lp Theorem
• Calderón–Zygmund operators are of weak type (1,1) Theorem
• Convergence in L¹(mu) implies convergence in measure Theorem
• De Giorgi local boundedness of homogeneous subsolutions Theorem
• Fatou limits for Poisson extensions of L1 boundary data Theorem
• Lebesgue differentiation theorem on ℝⁿ Theorem
• Maximal truncations: weak (1,1) and strong Lp bounds Theorem
• The Følner criterion for locally compact groups Theorem
• The Hardy-Littlewood maximal operator characterises Aₚ Theorem
• Weak maximum principle for coercive divergence-form equations Theorem
• Weighted L-p bounds for standard Calderon-Zygmund maximal truncations Theorem

Dependency tree · two levels

13 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

• John K. Hunter, Measure Theory Notes, ch. 4 (standard reference, not scraped)