Finite number of points in a probability
By Jessica Wood •
Let $(\Omega, \mathcal{A}, \mathbb{P})$ be a probability space, where $\mathcal{A}$ is the sigma field of all subsets of $\Omega$ and $\mathbb{P}$ is a probability function that assigns probability $p > 0$ to each point set of $\Omega$. Show that $\Omega$ must have a finite number of points.
I need show that $\Omega$ can't have more than $1/p$ points, but I don't know how to prove.
$\endgroup$ 11 Answer
$\begingroup$Assume $\Omega$ had more than $1/p$ points. Take distinct points $x_1, ..., x_n$ for $n > 1/p.$ Then set $S = \{x_1, ..., x_n\}.$ We have that$$\mathbb{P}(S) = \sum_{i=1}^n \mathbb{P}(x_i) = \sum_{i=1}^n p = pn > 1,$$which is absurd since no event has probability larger than 1.
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