A net $(x_α)_{α\in Γ}$ in a vector lattice $X$ is unbounded order convergent (uo-convergent) to $x$ if $|x_α-x| \wedge y \xrightarrow{o} 0$ for each $y \in X_+$, and is unbounded order Cauchy (uo-Cauchy) if the net $(x_α-x_{α'})_{Γ\times Γ}$ is uo-convergent to 0. In the first part of this article, we study uo-convergent and uo-Cauchy nets in Banach lattices and use them to characterize Banach lattices with the positive Schur property and KB-spaces. In the second part, we use the concept of uo-Cauchy sequences to extend Doob's submartingale convergence theorems to a measure-free setting. Our results imply, in particular, that every norm bounded submartingale in $L_1(Ω;F)$ is almost surely uo-Cauchy in $F$, where $F$ is an order continuous Banach lattice with a weak unit.
Several recent papers investigated unbounded convergences in Banach lattices. Combine all unbounded convergences, including unbounded order (norm, absolute weak, absolute weak*) convergence, we characterize L-weakly compact sets, L-weakly compact operators and M-weakly compact operators on Banach lattices. Some related results are obtained as well.
Several recent papers investigated unbounded versions of order and norm convergences in Banach lattices. In this paper, we study the unbounded variant of weak convergence and its relationship with other convergences. In particular, we characterize order continuous Banach lattices and reflexive Banach lattices in terms of this convergence.
The statistically unbounded $p$-convergence is an abstraction of the statistical order, unbounded order, and $p$-convergences. We investigate the concept of the statistically unbounded convergence on lattice-normed Riesz spaces with respect to statistical p-decreasing sequences. Also, we get some relations between this concept and the other kinds of statistical convergences on Riesz spaces.
Several recent papers investigated unbounded and statistical versions of order convergence and topology convergence in locally solid Riesz space. In this papers, we study the statistical unbounded order and topology convergence in Riesz spaces and Banach lattices. In particular, we study the relationship of those convergence and characterize order continuous, KB, reflexive Banach lattices in terms of these convergence.
The Fock transform recently introduced by the authors in a previous paper is applied to investigate convergence of generalized functional sequences of a discrete-time normal martingale $M$. A necessary and sufficient condition in terms of the Fock transform is obtained for such a sequence to be strong convergent. A type of generalized martingales associated with $M$ are introduced and their convergence theorems are established. Some applications are also shown.
Suppose $X$ is a vector lattice and there is a notion of convergence $x_α \rightarrow x$ in $X$. Then we can speak of an "unbounded" version of this convergence by saying that $(x_α)$ unbounded converges to $x\in X$ if $\lvert x_α-x\rvert \wedge u\rightarrow 0$ for every $u \in X_+$. In the literature, the unbounded versions of the norm, order and absolute weak convergence have been studied. Here we create a general theory of unbounded convergence but with a focus on uo-convergence and those convergences deriving from locally solid topologies.
Recently, the functionals different types of unbounded convergences (uo, un, uaw, uaw*) in Banach lattices were studied. In this paper, we study the continuous operators with respect to unbounded convergences. We first investigate the approximation property of continuous operators for unbounded convergence. Then we show some characterizations of the continuity of the continuous operators for uo, un, uaw and uaw*-convergence. Based on these results, we discuss the order-weakly compact operators on Banach lattices. Some related results are obtained as well.
Recently, the different types of unbounded convergences (uo, un, uaw, uaw*) in Banach lattices were studied. In this paper, we study the continuous functionals with respect to unbounded convergences. We first characterize the continuity of linear functionals for these convergences. Then we define the corresponding unbounded dual spaces and get their exact form. Based on these results, we discuss order continuity and reflexivity of Banach lattices. Some related results are obtained as well.
We characterize vector lattices in which unbounded order convergence is eventually order bounded. Among other things, the characterization provides a solution to \cite[Probl.23]{Az}.
The notion of almost everywhere convergence has been generalized to vector lattices as unbounded order convergence, which proves a very useful tool in the theory of vector and Banach lattices. In this short note, we establish some new results on unbounded order convergence that tie up some loose ends. In particular, we show that every norm bounded positive increasing net in an order continuous Banach lattice is uo-Cauchy and that every uo-Cauchy net in an order continuous Banach lattice has a uo-limit in the universal completion.
Based on the concept of unbounded absolutely weakly convergence, we give new characterizations of L-weakly compact sets. As applications, we find some properties of order weakly compact operators. Also, a new characterizations of order continuous Banach lattices are obtained.
A net $x_α$ in a lattice-normed vector lattice $(X,p,E)$ is unbounded $p$-convergent to $x\in X$ if $p(|x_α-x|\wedge u)\xrightarrow{o} 0$ for every $u\in X_+$. This convergence has been investigated recently for $(X,p,E)=(X,\lvert\cdot \rvert,X)$ under the name of $uo$-convergence, for $(X,p,E)=(X,\lVert\cdot\rVert,{\mathbb R})$ under the name of $un$-convergence, and also for $(X,p,{\mathbb R}^{X^*})$, where $p(x)[f]:=|f|(|x|)$, under the name $uaw$-convergence. In this paper we study general properties of the unbounded $p$-convergence.
As a generalization of almost everywhere convergence to vector lattices, unbounded order convergence has garnered much attention. The concept of boundedly uo-complete Banach lattices was introduced by N. Gao and F. Xanthos, and has been studied in recent papers by D. Leung, V.G. Troitsky, and the aforementioned authors. We will prove that a Banach lattice is boundedly uo-complete iff it is monotonically complete. Afterwards, we study completeness-type properties of minimal topologies; minimal topologies are exactly the Hausdorff locally solid topologies in which uo-convergence implies topologi…
Let $(x_α)$ be a net in a vector lattice normed by locally solid lattice $(X,p,E_τ)$. We say that $(x_α)$ is unbounded $p_τ$-convergent to $x\in X$ if $p(\lvert x_α-x\rvert\wedge u)\xrightarrowτ 0$ for every $u\in X_+$. This convergence has been studied recently for lattice-normed vector lattices as the $up$-convergence in \cite{AGG,AEEM,AEEM2}, the $uo$-convergence in \cite{GTX}, and, as the $un$-convergence in \cite{DOT,GX,GTX,KMT,Tr2}. In this paper, we study the general properties of the unbounded $p_τ$-convergence.
A net $(x_α)$ in a vector lattice $X$ is unbounded order convergent to $x \in X$ if $\lvert x_α- x\rvert \wedge u$ converges to $0$ in order for all $u\in X_+$. This convergence has been investigated and applied in several recent papers by Gao et al. It may be viewed as a generalization of almost everywhere convergence to general vector lattices. In this paper, we study a variation of this convergence for Banach lattices. A net $(x_α)$ in a Banach lattice $X$ is unbounded norm convergent to $x$ if $\lVert\lvert x_α- x\rvert \wedge u\rVert\to 0$ for all $u\in X_+$. We show that this convergence…
Unbounded order convergence has lately been systematically studied as a generalization of almost everywhere convergence to the abstract setting of vector and Banach lattices. This paper presents a duality theory for unbounded order convergence. We define the unbounded order dual (or uo-dual) $X_{uo}^\sim$ of a Banach lattice $X$ and identify it as the order continuous part of the order continuous dual $X_n^\sim$. The result allows us to characterize the Banach lattices that have order continuous preduals and to show that an order continuous predual is unique when it exists. Applications to the…
Using the spectral measure $μ_\mathbb{S}$ of the stopping time $\mathbb{S},$ we define the stopping element $X_\mathbb{S}$ as a Daniell integral $\int X_t\,dμ_\mathbb{S}$ for an adapted stochastic process $(X_t)_{t\in J}$ that is a Daniell summable vector-valued function. This is an extension of the definition previously given for right-order-continuous sub-martingales with the Doob-Meyer decomposition property. The more general definition of $X_\mathbb{S}$ necessitates a new proof of Doob's optional sampling theorem, because the definition given earlier for sub-martingales implicitly used Doo…
A net $(x_γ)_{γ\inΓ}$ in a locally solid Riesz space $(X,τ)$ is said to be unbounded $τ$-convergent to $x$ if $|x_γ-x|\wedge u\mathop{\oversetτ{\longrightarrow}} 0$ for all $u\in X_+$. We recall that there is a locally solid linear topology $\mathfrak{u}τ$ on $X$ such that unbounded $τ$-convergence coincides with $\mathfrak{u}τ$-convergence, and moreover, $\mathfrak{u}τ$ is characterised as the weakest locally solid linear topology which coincides with $τ$ on all order bounded subsets. It is with this motivation that we introduce, for a uniform lattice $(L,u)$, the weakest lattice uniformity $…
Let $x_α$ be a net in a locally solid vector lattice $(X,τ)$; we say that $x_α$ is unbounded $τ$-convergent to a vector $x\in X$ if $\lvert x_α-x \rvert\wedge w \xrightarrowτ 0$ for all $w\in X_+$. In this paper, we study general properties of unbounded $τ$-convergence (shortly, $uτ$-convergence). $uτ$-Convergence generalizes unbounded norm convergence and unbounded absolute weak convergence in normed lattices that have been investigated recently. Besides, we introduce $uτ$-topology and study briefly metrizabililty and completeness of this topology.
A sequence $(x_n)$ in a locally solid Riesz space $(E,τ)$ is said to be statistically unbounded $τ$-convergent to $x\in E$ if, for every zero neighborhood $U$, $\frac{1}{n}\big\lvert\{k\leq n:\lvert x_k-x\rvert\wedge u\notin U\}\big\rvert\to 0$ as $n\to\infty$. In this paper, we introduce this concept and give the notions $st$-$u_τ$-closed subset, $st$-$u_τ$-Cauchy sequence, $st$-$u_τ$-continuous and $st$-$u_τ$-complete locally solid vector lattice. Also, we give some relations between the order convergence and the $st$-$u_τ$-convergence.
Let $X$ be an ordered vector space. The net $\{x_α\}\subseteq X$ is semi unbounded order convergent to $x$ (in symbol $x_α\xrightarrow{suo}x$), if there is a net $\{y_β\}$, possibly over a different index set, such that $y_β\downarrow 0$ and for every $β$ there exists $α_0$ such that $\{\{\pm(x_α- x)\}^u,y\}^l\subseteq \{y_β\}^l$, whenever $α\geq α_0$ and for all $0\leq y \in X$. In vector lattice $E$, semi unbounded order convergence is equivalent with unbounded order convergence. We study some properties of this convergence and some of its relationships with others known order convergence.
In this paper, we investigate more about relationship between $uaw$ -convergence (resp. $un$-convergence) and the weak convergence. More precisely, we characterize Banach lattices on which every weak null sequence is $uaw$-null. Also, we characterize order continuous Banach lattices under which every norm bounded $un$-null net (resp. sequence) is weakly null. As a consequence, we study relationship between sequentially $uaw$-compact operators and weakly compact operators. Also, it is proved that every continuous operator, from a Banach lattice $E$ into a non-zero Banach space $X$, is unbounded…
In 1987, I. Labuda proved a general representation theorem that, as a special case, shows that the topology of local convergence in measure is the minimal topology on Orlicz spaces and $L_{\infty}$. Minimal topologies connect with the recent, and actively studied, subject of "unbounded convergences". In fact, a Hausdorff locally solid topology $τ$ on a vector lattice $X$ is minimal iff it is Lebesgue and the $τ$ and unbounded $τ$-topologies agree. In this paper, we study metrizability, submetrizability, and local boundedness of the unbounded topology, $uτ$, associated to $τ$ on $X$. Regarding …
A net $(x_α)$ in a vector lattice $X$ is said to be {unbounded order convergent} (or uo-convergent, for short) to $x\in X$ if the net $(\abs{x_α-x}\wedge y)$ converges to 0 in order for all $y\in X_+$. In this paper, we study unbounded order convergence in dual spaces of Banach lattices. Let $X$ be a Banach lattice. We prove that every norm bounded uo-convergent net in $X^*$ is $w^*$-convergent iff $X$ has order continuous norm, and that every $w^*$-convergent net in $X^*$ is uo-convergent iff $X$ is atomic with order continuous norm. We also characterize among $σ$-order complete Banach lattic…
Motivated by the equivalent definition of a continuous operator between Banach spaces in terms of weakly null nets, we introduce unbounded continuous operators by replacing weak convergence with the unbounded absolutely weak convergence ( $uaw$-convergence) in the definition of a continuous operator between Banach lattices. We characterize order continuous Banach lattices and reflexive Banach lattices in terms of these spaces of operators. Moreover, motivated by characterizing of a reflexive Banach lattice in terms of unbounded absolutely weakly Cauchy sequences, we consider pre-unbounded oper…
In this paper, we generalize the concept of unbounded norm (un) convergence: let $X$ be a normed lattice and $Y$ a vector lattice such that $X$ is an order dense ideal in $Y$; we say that a net $(y_α)$ un-converges to $y$ in $Y$ with respect to $X$ if $\Bigl\lVert\lvert y_α-y\rvert \wedge x\Bigr\rVert\to 0$ for every $x\in X_+$. We extend several known results about un-convergence and un-topology to this new setting. We consider the special case when $Y$ is the universal completion of $X$. If $Y=L_0(μ)$, the space of all $μ$-measurable functions, and $X$ is an order continuous Banach function …
