Almost uniform convergence

1993 ◽  
Vol 26 (1) ◽  
pp. 77-84 ◽  
Author(s):  
J. Ewert
Keyword(s):  
Uniform Convergence ◽  
Almost Uniform Convergence

scholarly journals Noncommutative strong maximals and almost uniform convergence in several directions – Addendum

2021 ◽  
Vol 9 ◽  
Author(s):  
José M. Conde-Alonso ◽  
Adrián M. González-Pérez ◽  
Javier Parcet
Keyword(s):  
Uniform Convergence ◽  
Almost Uniform Convergence

On Almost Uniform Convergence of Families of Functions

1964 ◽  
Vol 7 (1) ◽  
pp. 45-48 ◽  
Author(s):  
Elias Zakon
Keyword(s):  
Uniform Convergence ◽  
Present Note ◽  
Measure Zero ◽  
Slight Modification ◽  
Real Parameter ◽  
General Proof ◽  
Almost Uniform Convergence

In [5] Tolstov showed by a counterexample that Egoroff' s theorem on almost uniform convergence cannot be extended to families of functions (ft(x)}, with t a continuous real parameter. However, Frumkin [2] proved that this is possible provided that some sets of measure zero (depending on t) are disregarded when each particular ft(x) is considered. This interesting result was obtained by using the rather involvec machinery of Kantorovitch' s semi-ordered spaces and Lp spaces. In the present note we intend to give a simpler and more general proof. Indeed, it will be seen that only a slight modification of the standard proof of Egoroff's theorem is necessary to obtain Frumkin' s theorem in a more general form. We shall establish the following result.

scholarly journals A note on convergence in measure and selection principles

Filomat ◽  
2012 ◽  
Vol 26 (3) ◽  
pp. 473-477
Author(s):  
Dragan Djurcic ◽  
Ljubisa Kocinac
Keyword(s):  
Uniform Convergence ◽  
Almost Everywhere Convergence ◽  
Convergence In Measure ◽  
Mean Convergence ◽  
Measurable Functions ◽  
Almost Everywhere ◽  
Selection Principles ◽  
Almost Uniform Convergence ◽  
Special Modes

It is proved that some classes of sequences of measurable functions satisfy certain selection principles related to special modes of convergence (convergence in measure, almost everywhere convergence, almost uniform convergence, mean convergence).

scholarly journals Noncommutative strong maximals and almost uniform convergence in several directions

2020 ◽  
Vol 8 ◽  
Author(s):  
José M. Conde-Alonso ◽  
Adrián M. González-Pérez ◽  
Javier Parcet
Keyword(s):  
Initial Data ◽  
Uniform Convergence ◽  
Free Group ◽  
Group Algebra ◽  
Orlicz Spaces ◽  
Classical Inequality ◽  
Almost Everywhere ◽  
Poisson Semigroup ◽  
Original Argument ◽  
Almost Uniform Convergence

Abstract Our first result is a noncommutative form of the Jessen-Marcinkiewicz-Zygmund theorem for the maximal limit of multiparametric martingales or ergodic means. It implies bilateral almost uniform convergence (a noncommutative analogue of almost everywhere convergence) with initial data in the expected Orlicz spaces. A key ingredient is the introduction of the $L_p$ -norm of the $\limsup $ of a sequence of operators as a localized version of a $\ell _\infty /c_0$ -valued $L_p$ -space. In particular, our main result gives a strong $L_1$ -estimate for the $\limsup $ —as opposed to the usual weak $L_{1,\infty }$ -estimate for the $\mathop {\mathrm {sup}}\limits $ —with interesting consequences for the free group algebra. Let $\mathcal{L} \mathbf{F} _2$ denote the free group algebra with $2$ generators, and consider the free Poisson semigroup generated by the usual length function. It is an open problem to determine the largest class inside $L_1(\mathcal{L} \mathbf{F} _2)$ for which the free Poisson semigroup converges to the initial data. Currently, the best known result is $L \log ^2 L(\mathcal{L} \mathbf{F} _2)$ . We improve this result by adding to it the operators in $L_1(\mathcal{L} \mathbf{F} _2)$ spanned by words without signs changes. Contrary to other related results in the literature, this set grows exponentially with length. The proof relies on our estimates for the noncommutative $\limsup $ together with new transference techniques. We also establish a noncommutative form of Córdoba/Feffermann/Guzmán inequality for the strong maximal: more precisely, a weak $(\Phi ,\Phi )$ inequality—as opposed to weak $(\Phi ,1)$ —for noncommutative multiparametric martingales and $\Phi (s) = s (1 + \log _+ s)^{2 + \varepsilon }$ . This logarithmic power is an $\varepsilon $ -perturbation of the expected optimal one. The proof combines a refinement of Cuculescu’s construction with a quantum probabilistic interpretation of M. de Guzmán’s original argument. The commutative form of our argument gives the simplest known proof of this classical inequality. A few interesting consequences are derived for Cuculescu’s projections.

Sign in / Sign up

Export Citation Format

Share Document