Show that the collection of all subsets of a finite sample space satisfies the 3 conditions to be called a collection of events.

Question

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,

1. $S$ must be an event.

2. If $a \in S$ is an event, then so is $a^c$.

3. If $a_1, ..., a_n$ is a countable collection of events, then $\bigcup_{i=1}^{\infty} a_i$ is also an event.

Answer 1

Much of the post is not on the right track. The part that is on the right track is the list of three conditions. You need to show that the collection of all subsets of a (finite) set satisfies these conditions. The verifications will be in each case very short.

For example, is $S$ in our collection? Sure, every subset of $S$ is in our collection, and $S$ is a subset of itself.