The Egoroff theorem for non-additive measures in Riesz spaces (Q853396)

For a \(\sigma\)-algebra \(\mathcal{F}\) on a set \(X\) and a Riesz space \(V\), an increasing mapping \(\mu: \mathcal{F} \to V\), with \( \mu(\emptyset) =0\) is called a non-additive measure. \(\mu\) is called continuous from below if \( A_{n} \downarrow A \) implies \(\mu( A_{n}) \downarrow \mu( A)\), and continuous from above if \( A_{n} \uparrow A \) implies \(\mu( A_{n}) \uparrow \mu( A)\). If \(V= \mathbb{R}\), then it is known that if a non-additive measure \(\mu\) has continuity from above and below, then Egoroff's theorem holds. The author puts some conditions on \(V\) so that Egoroff's theorem may hold. The definition of \(V\) having asymptotic Egoroff's property: For \(m \in N\) and \(u \in V^{+}\), let \( u^{(m)} = \{ (u_{n_{1}, \dots, n_{m}}): (n_{1}, \dots, n_{m}) \in \mathbb{N}^{m} \} \subset V\). \( u^{(m)}\) is called \(u\)-multiple regulator if for every \(m\in \mathbb{N}\) and \( (n_{1}, \dots, n_{m}) \in \mathbb{N}^{m}\), \( u^{(m)}\) satisfies the conditions: (i) \( 0 \leq u_{n_{1}} \leq u_{n_{1}, n_{2}} \leq \dots u_{n_{1}, \dots, n_{m}} \leq u\), (ii) as \( n \to \infty\), \(u_{n} \downarrow 0, \; u_{n_{1}, n} \downarrow u_{n_{1}}, \dots, u_{n_{1}, \dots, n_{m}, n} \downarrow u_{n_{1}, \dots, n_{m}}\). \(V\) is said to have asymptotic Egoroff's property if for each \( u \in V^{+}\) and \(u\)-multiple regulator \(u^{(m)}\), we have (i) \( u_{\theta} = \sup_{m \in \mathbb{N}} u_{\theta(1), \dots, \theta(m)}\) exists for each \( \theta \in \Theta \) (here \(\Theta = \mathbb{N}^{\mathbb{N}})\), (ii) \(\inf_{\theta \in \Theta} u_{\theta} =0\). The main result is that if a Riesz space \(V\) has asymptotic Egoroff's property and \(\mu: \mathcal{F} \to V\) is non-additive and continuous from above and below, then Egoroff's theorem is valid for \(\mu\). Some other related results are proved. Also some examples of Riesz spaces, having asymptotic Egoroff's property and not having asymptotic Egoroff's property, are given.