Horizontal Egorov property of Riesz spaces

Abstract

We say that a Riesz space E E has the horizontal Egorov property if for every net ( f α ) (f_\alpha ) in E E , order convergent to f ∈ E f \in E with | f α | + | f | ≤ e ∈ E + |f_\alpha | + |f| \le e \in E^+ for all α \alpha , there exists a net ( e β ) (e_\beta ) of fragments of e e laterally convergent to e e such that for every β \beta , the net ( | f − f α | ∧ e β ) α \bigl (|f - f_\alpha | \wedge e_\beta \bigr )_\alpha e e -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 E E is said to satisfy the weak distributive law if for every e ∈ E + ∖ { 0 } e \in E^+ \setminus \{0\} the Boolean algebra F e \mathfrak {F}_e of fragments of e e satisfies the weak distributive law; that is, whenever ( Π n ) n ∈ N (\Pi _n)_{n \in \mathbb N} is a sequence of partitions of F e \mathfrak {F}_e , there is a partition Π \Pi of F e \mathfrak {F}_e such that every element of Π \Pi is finitely covered by each of Π n \Pi _n (e.g., every measurable Boolean algebra is so). Using a new technical tool, we show that for every net ( f α ) (f_\alpha ) order convergent to f f in a Riesz space with the horizontal Egorov property there are a horizontally vanishing net ( v β ) (v_\beta ) and a net ( u α , β ) ( α , β ) ∈ A × B (u_{\alpha , \beta })_{(\alpha , \beta ) \in A \times B} , which uniformly tends to zero for every fixed β \beta such that | f − f α | ≤ u α , β + v β |f - f_\alpha | \le u_{\alpha , \beta } + v_\beta for all α , β \alpha , \beta .