What is a Sigma-Algebra? Definition and Examples

Summary
This class discusses the importance of sigma-algebra in probability theory. Sigma-algebra is a structure that contains all measurable events in a sample space, enabling the definition of a probability measure. Practical examples, such as coin tosses and the lifespan of an electronic device, explain how sigma-algebra is constructed from the parts of a sample space. The Borel sigma-algebra, associated with a continuous sample space, is also presented, and its Borel events are explained.

LEARNING OBJECTIVES:
Upon completion of this class, the student will be able to:

Understand the definition and characteristics of a Sigma-Algebra, as a mathematical structure that enables the definition of a probability measure.
Identify the elements that make up a Sigma-Algebra and its relationship with measurable events in a sample space.

TABLE OF CONTENTS
DEFINITION OF A SIGMA-ALGEBRA
SIGMA-ALGEBRA IN COIN TOSSING
SIGMA-ALGEBRAS IN CONTINUOUS CASES

Measurable events appear in the probability space through the sigma-algebra. Through this idea, an initially intuitive notion is transformed into a formally mathematical structure that allows defining a probability measure.

Definition of a Sigma-Algebra

A Sigma-Algebra $\Sigma$ (or σ-algebra) is a structure that contains all measurable events of a sample space. The pair $\Sigma_{\Omega} = (\Omega, \mathcal{A}_{\Omega})$ is said to be a σ-algebra of a sample space $\Omega$ if it satisfies the following:

$\emptyset,\Omega \in \mathcal{A}_\Omega$
$\left(E \in \mathcal{A}_\Omega \right) \rightarrow (E^c = \Omega\setminus E \in \mathcal{A}_\Omega)$
$\left(E_1, E_2 \in \mathcal{A}_\Omega \right) \rightarrow (E_1 \cup E_2 \in \mathcal{A}_\Omega)$

All objects $E\in\mathcal{A}_\Omega$ are called Events of $\Omega$ .

The sigma-algebra in coin tosses

EXAMPLE 1

For a coin toss, the σ-algebra is given by

\Sigma_{1m}=(\Omega_{1m}, \mathcal{A}_{1m})

, where

$\Omega_{1m}= \{H,T\}$
$\mathcal{A}_{1m}= \{\emptyset,\{H\},\{T\}, \Omega_{1m}\}$

Each element of $\mathcal{A}_{1m}$ is an event identified as follows:

$\emptyset$ “Neither heads nor tails appear” (it is the impossible event).
$\{H\}$ “It is the event in which heads appear”.
$\{T\}$ “It is the event in which tails appear”.
$\Omega_{1m}= \{H,T\}$ “Either heads or tails appears” (it is the certain event).

EXAMPLE 2

If we toss two coins instead of one, a possible σ-algebra

\Sigma_{2m}=(\Omega_{2m}, \mathcal{A}_{2m})

can be obtained from the parts of

\Omega_{2m}

. Thus we have the following:

$\Omega_{2m}= \{(C,C);(C,S); (S,C); (S,S)\}$
$\mathcal{A}_{2m}=\mathcal{P}(\Omega_{2m}) = \cdots$ $\cdots = \{\emptyset; \{(C,C)\}; \left\{(C,S)\}; \{(S,C)\}; \{(S,S)\}; \cdots \right.$ $\cdots; \{(C,C);(C,S)\};\{(C,C);(S,C)\};\{(C,C);(S,S)\};\cdots$ $\cdots; \{(C,S);(S,C)\};\{(C,S);(S,S)\};\{(S,C);(S,S)\};\cdots$ $\cdots; \{(C,C);(C,S);(S,C)\};\{(C,C);(C,S);(S,S)\}\cdots$ $\left. \cdots; {(C,C);(S,C);(S,S)}; {(C,S);(S,C);(S,S)}; \Omega_{2m}\right\}$

Each element of $\mathcal{A}{2m}$ is an event of $\Omega{2m}$ . Here are some of them:

$\emptyset$ “No results occur” (the impossible event).
$\{(C,C)\}$ “Two consecutive heads”.
$\{(C,S)\}$ “Head followed by tail”.
$\vdots$
$\{(C,C);(C,S)\}$ “First coin shows heads, second coin can show any result”.
$\{(C,C);(S,C)\}$ “First coin can show any result, second coin shows heads”.
$\{(C,C);(S,S)\}$ “Both tosses have the same result”.
$\vdots$
$\{(C,C);(C,S);(S,S)\}$ “If the first toss shows tails, then the second toss also shows tails; otherwise, the second toss can show any result”.
$\{(C,C);(S,C);(S,S)\}$ “If the first toss shows heads, then the second toss also shows heads; otherwise, the second toss can show any result”.
$\vdots$
$\Omega_{2m}$ “Any possible result can occur” (the certain event).

Sigma-algebras in Continuous Cases

EXAMPLE 3

For the lifetime (measured in hours) of an electronic device that could fail at any moment, the σ-algebra

\Sigma_e = (\Omega_e, \mathcal{A}_e)

is given by

$\Omega_e = [0, \infty[$
$\mathcal{A}_e = \{I \; | \; I \subseteq \Omega_e \}$

Thus, the intervals $I_t = ]0,t[\in\mathcal{A}_e$ can be interpreted as “the electronic device works correctly for an interval of $t$ consecutive hours until it fails.

The probability σ-algebra associated with a continuous sample space is also known as the Borel σ-algebra and its events are known as Borel sets.

What is a Sigma-Algebra? Definitions and Examples

What is a Sigma-Algebra? Definition and Examples

Definition of a Sigma-Algebra

The sigma-algebra in coin tosses

Sigma-algebras in Continuous Cases

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