If we assign probabilities to every elementary event such that they sum up to 1 and they're all positive, can we say that this function is a probability function with its domain equal to the set of all subsets of the union of those elementary events? Or should we also specify that P(A or B) = P(A) + P(B) for all A, B where A and B=Null set?
Edit:
$\Omega = \{\omega_1,\omega_2,\omega_3\}$
Let $f:\Omega \rightarrow[0,1]$ such that:
$f(\omega_1)=0.2$, $f(\omega_2)=0.3$, $f(\omega_3)=0.5$.
Let $F:2^\Omega\rightarrow[0,1]$ such that:
$F(\{\omega_i\})=f(\omega_i), i\in \{1,2,3\}$,
$F(\{\omega_1,\omega_2\})=f(\omega_1)+f(\omega_2)$
$F(\{\omega_1,\omega_3\})=f(\omega_1)+f(\omega_3)$
$F(\{\omega_2,\omega_3\})=f(\omega_2)+f(\omega_3)$
$F(\{\omega_1,\omega_2\,\omega_3\})=f(\omega_1)+f(\omega_2)+f(\omega_3)$
$F(\emptyset)=0$
Then we can say that F is a probability function, right? Can we say that this is the unique probability function generated by f (not speaking formally)?