Horizontal Egorov property of Riesz spaces,Proceedings of the American Mathematical Society

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Horizontal Egorov property of Riesz spaces
Proceedings of the American Mathematical Society ( IF 1 ) Pub Date : 2020-10-20 , DOI: 10.1090/proc/15235
Mikhail Popov

Abstract:We say that a Riesz space

has the horizontal Egorov property if for every net $(f_\alpha )$ in

, order convergent to $f \in E$ with $\vert f_\alpha \vert + \vert f\vert \le e \in E^+$ for all $\alpha$ , there exists a net $(e_\beta )$ of fragments of

laterally convergent to

such that for every $\beta$ , the net $\bigl (\vert f - f_\alpha \vert \wedge e_\beta \bigr )_\alpha$

-uniformly tends to zero. Our main result asserts that every Dedekind complete Riesz space which satisfies the weak distributive law possesses the horizontal Egorov property. A Riesz space

is said to satisfy the weak distributive law if for every $e \in E^+ \setminus \{0\}$ the Boolean algebra $\mathfrak{F}_e$ of fragments of

satisfies the weak distributive law; that is, whenever $(\Pi _n)_{n \in \mathbb{N}}$ is a sequence of partitions of $\mathfrak{F}_e$ , there is a partition $\Pi$ of $\mathfrak{F}_e$ such that every element of $\Pi$ is finitely covered by each of $\Pi _n$ (e.g., every measurable Boolean algebra is so). Using a new technical tool, we show that for every net $(f_\alpha )$ order convergent to

in a Riesz space with the horizontal Egorov property there are a horizontally vanishing net $(v_\beta )$ and a net $(u_{\alpha , \beta })_{(\alpha , \beta ) \in A \times B}$ , which uniformly tends to zero for every fixed $\beta$ such that $\vert f - f_\alpha \vert \le u_{\alpha , \beta } + v_\beta$ for all $\alpha , \beta$ .

中文翻译：

Riesz空间的水平Egorov属性

摘要：我们说里斯空间

具有水平埃格罗夫财产，如果为每一个网中，为了收敛到与所有的，存在一个净的碎片横向收敛到这样，对于每一个，净-uniformly趋向于零。我们的主要结果认为，每个满足弱分布定律的Dedekind完全Riesz空间都具有水平的Egorov属性。如果对于每个满足布尔碎片代数的布尔代数都满足弱分布定律，则称Riesz空间满足弱分布定律；也就是说，只要是的分区序列，就会有一个分区 $（f_ \ alpha）$

$f \ in E$ $\ vert f_ \ alpha \ vert + \ vert f \ vert \ le e \ in E ^ +$ $\ alpha$ $（e_ \ beta）$

$\ beta$ $\ bigl（\ vert f-f_ \ alpha \ vert \ wedge e_ \ beta \ bigr）_ \ alpha$

$e \ in E ^ + \ setminus \ {0 \}$ $\ mathfrak {F} _e$

$（\ Pi _n）_ {n \ in \ mathbb {N}}$ $\ mathfrak {F} _e$ $\ Pi$ 的，使得每个元件由每个是有限覆盖（例如，每可测量布尔代数是如此）。使用新的技术工具，我们表明，每网为了收敛到与水平埃格罗夫属性里斯空间有水平等于零的净和净，其一致地趋向于零每隔一定使得所有。 $\ mathfrak {F} _e$ $\ Pi$ $\ Pi _n$ $（f_ \ alpha）$

$（v_ \ beta）$ $（u _ {\ alpha，\ beta}）_ {（\ alpha，\ beta）\在A \ times B}$ $\ beta$ $\ vert f-f_ \ alpha \ vert \ le u _ {\ alpha，\ beta} + v_ \ beta$ $\ alpha，\ beta$

更新日期：2020-11-12

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