## The Egoroff theorem for non-additive measures in Riesz spaces.(English)Zbl 1106.28005

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.

### MSC:

 28B15 Set functions, measures and integrals with values in ordered spaces 28C20 Set functions and measures and integrals in infinite-dimensional spaces (Wiener measure, Gaussian measure, etc.)

### Keywords:

Egoroff’s theorem; asymptotic Egoroff’s property
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