Is this set of event axioms complete?

Question

In the notes' first chapter of the course "Discrete Stochastic Processes" presented by Prof. Robert Gallager, the "Axioms for events" defined as follows:

1.2.1 Axioms for events [Chapter 1: Introduction and review of probability, page 6]

Given a sample space $\Omega$ , the class of subsets of $\Omega$ that constitute the set of events satisfies the following axioms:

1. $\Omega$ is an event.
2. For every sequence of events $A_1, A_2, \ldots$, the union $\bigcup_{n=1}^{\infty} A_n$ is an event.
3. For every event $A$, the complement $A^c$ is an event.

The notes also states that:

Note that the axioms do not say that all subsets of $\Omega$ are events. In fact, there are many rather silly ways to define classes of events that obey the axioms.

Based on aforementioned axioms I cannot find any event except $\Omega$ and $\emptyset$.

For example, suppose we have $\Omega = \{1, 2, 3\}$, is $A=\{1\}$ an event? If answer is yes, based on which axioms?

Answer 1

A collection of events satisfying axioms 1-3 is called a $\sigma$-algebra. You, no doubt, encountered this in the context of probability space $$(\Omega,\mathcal{A},P)$$ where the second member of the tiple $\mathcal{A}$, the collection of events on which the probability measure $P$ is defined, must be a $\sigma$-algebra. Both $\{\emptyset, \Omega\}$ and $2^{\Omega}$, the set of all subsets of $\Omega$, are examples of $\sigma$-algebras. But, between these two extremes, there are many more $\sigma$-algebras, and you have to choose one of them to be your set of events, when you define a probability space.

When $\Omega$ is finite, you will usually choose $2^{\Omega}$ for your set of events. But, when $\Omega=\mathbb{R}$, it turns out that there is, for example, no probability measure defined on the whole $2^{\mathbb{R}}$ such that $P(\{x\})=0$ for every singleton $\{x\} \subset \mathbb{R}$, and if we are interested in such measures, we have to restrict our attention to some smaller $\sigma$-algebra of events, that will make it possible to define such a probability measure. In case of $\Omega=\mathbb{R}$ it will usually be the $\sigma$-algebra of Borel sets $\mathcal{B}(\mathbb{R})$.

In short, you choose the $\sigma$-algebra of measurable events so that it is small enough to enable us to define a desired probability measure on it, and large enough to encompass all the sets we are interested in. In principle, it will vary from probability space to probability space.

Answer 2

The axioms are right, and one cannot say that $A = \{1\}$ is an event.

It may be that your $\sigma$-algebra of events is simply $\{\emptyset, \Omega\}$.

So to answer the exercise, one should say: "Based on the axioms one cannot say whether $A$ is an event".