Claim:Giving probabilities for simple outcomes is enough

Claim (see proof):

If [math]\href{/cs2800/wiki/index.php/S}{S} [/math] is a finite sample space, and [math]P : \href{/cs2800/wiki/index.php/S}{S} → \href{/cs2800/wiki/index.php?title=%E2%84%9D&action=edit&redlink=1}{ℝ} [/math] is a function satisfying

1. [math]P(s) ≥ 0 [/math] for all [math]s \href{/cs2800/wiki/index.php/%E2%88%88}{∈} \href{/cs2800/wiki/index.php/S}{S} [/math]
2. [math]\sum_{s \href{/cs2800/wiki/index.php/%E2%88%88}{∈} \href{/cs2800/wiki/index.php/S}{S}} P(s) = 1 [/math]

then there is a unique probability measure [math]\href{/cs2800/wiki/index.php/Pr}{Pr} [/math] having [math]\href{/cs2800/wiki/index.php/Pr}{Pr}(\{s\}) = P(s) [/math] for all [math]s [/math].