Let $\left(\Omega, \mathcal{F}, \mathbb{P}\right)$ be a finite probability space equipped with a filtration, i.e an increasing sequence of $\sigma$-algebras included in $\mathcal{F}$ : $\mathcal{F}_0, \mathcal{F}_1, \ldots, \mathcal{F}_N$. Denote $\Gamma$ the set of all non-negative random variables X such that $\mathbb{P}$ (X > 0) > 0. Let K = $\{X \in \Gamma \mid \sum_{w} X\left(w\right) = 1\}$. Show that K is a convex compact set.
I know how to show that K is a convex set, and since I'm dealing with a finite probability space, so if I want to show that K is a compact set, I just need to show that K is a bounded and closed set. The problem is that I don't know how to show that K is a closed set. Can anyone help me to proof that?
Thank you.