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?

  • $\begingroup$ "the class of subsets of $\Omega$", so I would think that if $A$ is a subset of $\Omega$, then yes. $\endgroup$ – Anthony Aug 12 '15 at 13:21
  • $\begingroup$ So essentially you say every subset of $\Omega$ is an event? $\endgroup$ – Taher Rahgooy Aug 12 '15 at 13:22
  • $\begingroup$ Those conditions looks consistent but not complete. I.e. they look like they are necessary for any characterisation of events but not sufficient to define all events. $\endgroup$ – Colm Bhandal Aug 12 '15 at 13:25
  • $\begingroup$ As I understand it, $A$ needs to be in $\Omega$, and it also needs to be in the subset of $\Omega$ which represent the set of events. $\endgroup$ – Anthony Aug 12 '15 at 13:27
  • $\begingroup$ You are correct in that only $\Omega$ and $\emptyset$ can be deduced as being events from these axioms. In fact, from only axioms $1$ and $3$. $\endgroup$ – Colm Bhandal Aug 12 '15 at 13:27

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.

  • $\begingroup$ You can put a probability measure on $2^\Omega$ for uncountable $\Omega$. For instance take a countable subset, pick your favorite probability measure to put on the subset, and assign probability zero to any set disjoint from this subset. You can't put a translation-invariant measure on $2^{\mathbb{R}}$, but that's another story entirely. $\endgroup$ – Ian Aug 12 '15 at 14:25
  • $\begingroup$ @Ian Yes, thank you. I corrected my answer. $\endgroup$ – Zoran Loncarevic Aug 12 '15 at 15:50

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".


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.