As title. Suppose that the sample space $S$ of some experiment is finite. Show that the collection of all subsets of $S$ satisfies the three conditions required to be called the collection of events.
So sample space $S$ is finite, with $N$ possible outcomes, where $N$ is a positive integer. Thus $S$ has the form $S = \{w_1, ..., w_N\}$, where $w_i$ represents the $i$-th possible outcome, for $i \in \{1, ..., N\}$.
This sample space contains $2^N$ possible events like a power set.
We can construct a probability measure by defining probabilities $p_i$ for each particular outcome $w_i \in S$.
$P[w_i] = p_i, \forall i \in \{1, ..., N\}$
I was thinking of defining the probability measure to all events $A \subseteq S$ by defining $P(\emptyset)=0$, and $P(A)$ for each non-empty subset $A \subseteq S$ as follows:
$P(A) = \sum_{w_i \in A} P(w_i)$
where the summation on the right-hand side represents a sum over each of the elements $w_i$ in the set $A$.
My question is, am I on the right track, if so, how do I show that this definition satisfies the three axioms of probability? That is,
$S$ must be an event.
If $a \in S$ is an event, then so is $a^c$.
If $a_1, ..., a_n$ is a countable collection of events, then $\bigcup_{i=1}^{\infty} a_i$ is also an event.