A strong law of large numbers under sublinear expectations

We consider a sequence of independent and identically distributed (i.i.d.) random variables $ \{\xi_k\} $under a sublinear expectation $ \mathbb{E} = \sup_{P\in\Theta}E_P $. We first give a new proof to the fact that, under each $ P\in\Theta $, any cluster point of the empirical averages $ \bar{\xi}_n = (\xi_1+\cdots+\xi_n)/n $ lies in $ [\underline{\mu}, \overline{\mu}] $ with $ \underline{\mu} = -\mathbb{E}[-\xi_1], \overline{\mu} = \mathbb{E}[\xi_1] $. Next, we consider sublinear expectations on a Polish space $ \Omega $, and show that for each constant $ \mu\in [\underline{\mu},\overline{\mu}] $, there exists a probability $ P_{\mu}\in\Theta $ such that$ \lim\limits_{n\rightarrow \infty}\bar{\xi}_n = \mu, \; P_{\mu}\text{-a.s.}, $(0.1) supposing that $ \Theta $ is weakly compact and $ \{\xi_n\}\in L^1_{\mathbb{E}}(\Omega) $. Under the same conditions, we obtain a generalization of (0.1) in the product space $ \Omega = \mathbb{R}^{\mathbb{N}} $ with $ \mu\in [\underline{\mu},\overline{\mu}] $ replaced by $ \Pi = \pi(\xi_1, \cdots,\xi_d)\in [\underline{\mu},\overline{\mu}] $. Here $ \pi $ is a Borel measurable function on $ \mathbb{R}^d $, $ d\in\mathbb{N} $. Finally, we characterize the triviality of the tail $ \sigma $ -algebra of the i.i.d. random variables under a sublinear expectation.