Sequential suprema, infima, limsup, liminf, and pointwise limits of measurable functions are measurable

Depends on

• Limit superior and limit inferior of a real sequence as $\inf_n \sup_{k \ge n} x_k$ and $\sup_n \inf_{k \ge n} x_k$ in $\overline{\mathbb{R}}$
• A real sequence converges to $L \in \mathbb{R}$ iff $\liminf x_k = \limsup x_k = L$, and diverges to $\pm\infty$ iff both equal $\pm\infty$
• Threshold characterisations of real-valued and extended-real-valued measurability

Used by

• Almost-sure convergence of a random series Definition
• Conditional expectation for nonnegative variables Definition
• Measurable and decomposable operator fields Definition
• The weighted maximal function of a doubling weight Definition
• Upcrossing number of an interval Definition
• Wiener measure on continuous path space Definition
• A deterministic integral construction of a Gaussian process Example
• A Sierpinski gasket computed by hand Example
• A topological invariant mean yields norm-approximately invariant densities Lemma
• Adapted continuous processes are progressively measurable Lemma
• Borel sigma-algebra of continuous path space is generated by coordinates Lemma
• Cauchy sequences in probability have a measurable limit Lemma
• Hilbert cube has a bimeasurable real coding Lemma
• L² convolution on a compact group is Hilbert–Schmidt Lemma
• Measurable Gram-Schmidt and constant-field trivializations on dimension strata Lemma
• Measurable sections have measurable pointwise inner products Lemma
• Rational conditional distribution functions produce real regular kernels Lemma
• Regular conditional kernels factor through a standard borel conditioning variable Lemma
• Reiter functions can be cut down to Følner sets Lemma
• Simultaneous rational conditional distribution function versions Lemma
• Digit-position density determines Hausdorff dimension Proposition
• For sigma-finite measures, the section-measure functions are measurable Proposition
• A function measurable for a completion is almost everywhere equal to one measurable for the original sigma-algebra Theorem
• Birkhoff's theorem for an ergodic probability system Theorem
• Bochner dominated convergence theorem Theorem
• Bochner integrability criterion Theorem
• Cauchy sequences in measure converge in measure Theorem
• Chain rule for globally Lipschitz scalar maps of Sobolev functions Theorem
• Conditional fatou and dominated convergence Theorem
• Direct integrals of measurable Hilbert fields are Hilbert spaces Theorem
• Disintegration of a joint law on standard borel spaces Theorem
• L two kernels give Hilbert–Schmidt operators Theorem
• Levy continuity theorem converse Theorem
• Localized Ito integral Theorem
• Reverse martingale convergence Theorem
• Riesz-Fischer completeness of Lᵖ for 1 ≤ p ≤ ∞ Theorem
• Separable dual spaces have the Radon--Nikodym property Theorem
• Strong Markov property of Brownian motion Theorem
• Taking out what is known Theorem
• The ACL characterisation of W^1,p Theorem

…and 2 more results.

Dependency tree · two levels

27 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

• Sheldon Axler, Measure, Integration and Real Analysis, Proposition 2.53 (standard reference, not scraped)