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Probability space

by , PhD

A probability space is a triple [eq1], where Omega is a sample space, $	ciFourier $ is a sigma-algebra of events and $QTR{rm}{P}$ is a probability measure on $	ciFourier $.

Table of Contents

Elements of a probability space

The three building blocks of a probability space can be described as follows:

A simple example

Suppose that the probabilistic experiment consists in extracting a ball from an urn containing two balls, one red ($R$) and one blue ($B$).

The sample space is[eq2]

A possible sigma-algebra of events is[eq3]where $emptyset $ is the empty set.

The four events could be described as follows:

A possible probability measure $QTR{rm}{P}$ on $	ciFourier $ is[eq4]

Another possibility would be to define the so-called trivial sigma-algebra[eq5]and specify a probability measure $QTR{rm}{P}$ on [eq6] as[eq7]

Sample space

How are the three building blocks of a probability space defined?

The first one, the sample space Omega, is a primitive concept, loosely defined as the set of all possible outcomes of the probabilistic experiment.

The other two building blocks are instead defined rigorously, by enumerating the properties (or axioms) that they need to satisfy.


Let $	ciFourier $ be a set whose elements are subsets of Omega. Then, $	ciFourier $ is a sigma-algebra if and only if it satisfies the following axioms:

  1. [eq8];

  2. If $Ain 	ciFourier $, then [eq9] (where $A^{c}$ is the complement of A, also denoted by $Omega setminus A$)

  3. If [eq10] is a countable collection of elements of $	ciFourier $ and [eq11], then $Ain 	ciFourier $.

It is possible to prove that these axioms imply:

Probability measure

Let $QTR{rm}{P}$ be a function that associates a real number to each element of the sigma-algebra $	ciFourier $. Then, $QTR{rm}{P}$ is a probability measure if and only if it satisfies the following axioms:

  1. If $Ain 	ciFourier $, then [eq15];

  2. [eq16];

  3. If [eq13] is a countable collection of disjoint elements of $	ciFourier $ (i.e., [eq18] if $j
eq k$), then [eq19].

General explanation

Why do we impose all these axioms?

Basically, it is for historical reasons and mathematical convenience.

Before Andrey Kolmogorov used the axioms above to define probability, mathematicians had proposed other definitions of probability. Those definitions had several flaws, but they all implied that probability satisfies the axioms above.

By using the axioms as a definition, Kolmogorov was able to develop a theory of probabilities that was logically coherent and without mathematical flaws.

Since then, this definition has been productively used by generations of statisticians, who have built myriads of incredibly useful results upon Kolmogorov's theory.

If it is the first time that you see these axioms, the best strategy to approach them is to memorize them and then convince yourself that they are reasonable. The next sections will help you to do so.

Explanation of the axioms of a sigma-algebra

Let us start from sigma-algebras.

As we have said, the sigma-algebra contains all the subsets of Omega to which we wish to assign probabilities.

So, the first thing to keep in mind is that we do not necessarily need to assign a probability to each possible subset of Omega.

Example Suppose that the sample space is the unit interval:[eq20]Define[eq21]You can easily check that $	ciFourier $ is a sigma-algebra, by verifying that it satisfies the three axioms. It contains very few subsets of Omega. However, if we are interested in the probability of the event $left{ 1
ight} $, we do not need anything more complicated. Clearly, we can build more complex sigma-algebras. For example, the smallest sigma-algebra that contains the interval $left( 0,1
ight) $ is[eq22]A widely used sigma-algebra is the Borel one, which contains all the open subintervals of $left[ 0,1
ight] $.

Let us now turn to the three defining axioms.

The three axioms defining a sigma-algebra.

Axiom 1 means that we should always be allowed to speak of the trivial event "something will happen".

According to Axiom 2, if we take into consideration the possibility that one of the things in the set A will happen, then we must also consider the alternative possibility that none of the things in the set A will happen.

According to Axiom 3, if we contemplate the possibility that some events will happen (separately for each of them), then we must also be able to assess the possibility that at least one of them will happen (their union).

Axiom 3 is formulated for countable collections of events for mathematical convenience. However, it implies that an analogous property holds for finite collections of events.

In fact, by setting [eq23]we obtain[eq24]

Explanation of the axioms of a probability measure

As we have said, a probability measure attaches a real number (a probability) to each event in the sigma-algebra.

The three axioms defining a probability measure.

Axioms 1 and 2 are established arbitrarily: we simply decide, in line with a centuries-old tradition, that a probability is a positive number smaller than or equal to 1, and that the probability of the sure event is 1.

According to Axiom 3 (called countable additivity), the sum of the probabilities of some disjoint events must be equal to the probability that at least one of those events will happen (their union).

The countable additivity axiom is probably easier to interpret when we set [eq25]so as to obtain [eq26]which, for $n=2$, becomes[eq27]

More details and explanations

The lecture on the mathematics of probability contains a more detailed explanation of the concept of probability space and of the properties that must be satisfied by its building blocks.

Keep reading the glossary

Previous entry: Probability mass function

Next entry: Random matrix

How to cite

Please cite as:

Taboga, Marco (2021). "Probability space", Lectures on probability theory and mathematical statistics. Kindle Direct Publishing. Online appendix.

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