Fatou's lemma

Depends on

• Monotone convergence for the integral
• Monotonicity and nonnegative homogeneity of the nonnegative integral
• Closure properties of measurable functions used by the integral

Used by

• Reverse Fatou's lemma under an integrable majorant Corollary
• Strong convergence of subcritical powers Corollary
• Fatou can be strict and domination can fail simultaneously Counterexample
• Mass can escape to infinity under pointwise convergence Counterexample
• The Lusin area function for a fixed admissible kernel and aperture Definition
• A nonnegative martingale converges almost surely Example
• Integral of Brownian motion against itself Example
• Ito formula for Brownian powers Example
• FALSE: Fatou's lemma is always an equality False statement
• A self-adjoint operator generates a strongly continuous unitary group Lemma
• Blaschke factorization of a Nevanlinna-class function Lemma
• De Giorgi oscillation reduction: one half-level set is small Lemma
• Distributional Laplacian of a compact logarithmic potential Lemma
• Ergodic averages converge in Lp on finite-measure spaces Lemma
• Log-integrability of the boundary values of a Hardy function Lemma
• Poisson-Jensen inequality for Hardy functions Lemma
• Real L2 multipliers and unitary transport Lemma
• Reverse Holder from a distribution estimate for a doubling weight Lemma
• Separate holomorphy forces local boundedness on smaller polydiscs Lemma
• Sup-norm and first-derivative bounds by the L² norm on compact subsets Lemma
• The circle maximal function is weak type one one for finite measures Lemma
• The disc trace space is the Hardy boundary space and the Szegő family reproduces H² Lemma
• The half-space trace lies in the fractional Slobodeckij space Lemma
• The L1 group algebra has a unit exactly when the group is discrete Proposition
• For 0<p<1 the Hᵖ functional is a quasi-norm, and Hᵖ is a quasi-Banach space Remark
• A maximum principle for the Smirnov class: N^+∩ Lᵖ=Hᵖ Theorem
• Birkhoff pointwise ergodic theorem Theorem
• Boundary values and log-integrability of Nevanlinna-class functions Theorem
• Calderón–Zygmund operators are of weak type (1,1) Theorem
• Doob submartingale convergence theorem Theorem
• Fatou's boundary theorem for analytic Hardy spaces Theorem
• For a nondecreasing function, the derivative is measurable and integrable and its integral is bounded by the total increase Theorem
• Hitting probability as minimal harmonic extension Theorem
• Kolmogorov convergence criterion Theorem
• Maximal truncations: weak (1,1) and strong Lp bounds Theorem
• Meyers–Serrin density on an arbitrary open set Theorem
• Multidimensional Ito formula for Brownian-driven processes Theorem
• One-dimensional Ito formula Theorem
• Superharmonic majorants bound exit costs Theorem
• The ACL characterisation of W^1,p Theorem

…and 4 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, Theorem 7.8 (standard reference, not scraped)
• John K. Hunter, Measure Theory Notes, Theorem 4.22 (standard reference, not scraped)