Generated Sigma-Algebras

What Does “Generated” Mean?

The Examples of Sigma-Algebras tutorial showed several ways to build sigma-algebras. A different question is often more useful: given a collection of subsets of a space, what is the smallest sigma-algebra that contains them? The collection may not itself be a sigma-algebra, so it must be enlarged until it has the required closure properties. The result is called the sigma-algebra generated by that collection.

Let $X$ be a set and let $\mathcal{A}\subseteq\mathcal{P}(X)$ be a collection of subsets of $X$. We use the existence result from the Sigma-Algebras tutorial: there is a smallest sigma-algebra on $X$ containing $\mathcal{A}$. Its defining minimality is the key tool in working with generators.

Definition: The sigma-algebra generated by $\mathcal{A}$ on $X$, denoted $\sigma_X(\mathcal{A})$, is the smallest sigma-algebra on $X$ that contains every member of $\mathcal{A}$. When the underlying space is clear, we write $\sigma(\mathcal{A})$.

Equivalently, the generated sigma-algebra is the intersection of all sigma-algebras on $X$ that contain $\mathcal{A}$. This family is nonempty because $\mathcal{P}(X)$ is a sigma-algebra containing $\mathcal{A}$. The existence theorem ensures that the intersection gives the smallest such sigma-algebra. This description is useful conceptually, but it does not usually list the generated sets explicitly.

The Minimality Property

To show that a sigma-algebra contains $\sigma_X(\mathcal{A})$, it is enough to show that it contains the generators. Conversely, every member of $\sigma_X(\mathcal{A})$ must belong to any sigma-algebra that contains those generators. This simple equivalence is the universal property of generation.

Theorem (Universal Property of a Generated Sigma-Algebra): Let $\mathcal{A}\subseteq\mathcal{P}(X)$, and let $\mathcal{F}$ be a sigma-algebra on $X$. Then $$ \mathcal{A}\subseteq\mathcal{F} \quad\Longleftrightarrow\quad \sigma_X(\mathcal{A})\subseteq\mathcal{F}. $$

Proof. Suppose first that $\mathcal{A}\subseteq\mathcal{F}$. Then $\mathcal{F}$ is a sigma-algebra containing every generator in $\mathcal{A}$. By the minimality in the definition of $\sigma_X(\mathcal{A})$, we have $\sigma_X(\mathcal{A})\subseteq\mathcal{F}$.

For the reverse implication, every member of $\mathcal{A}$ belongs to $\sigma_X(\mathcal{A})$ by definition. Thus $\sigma_X(\mathcal{A})\subseteq\mathcal{F}$ implies $\mathcal{A}\subseteq\mathcal{F}$. Both directions hold. $\square$

One immediate consequence is a convenient test for equality. If two collections of generators each lie in the sigma-algebra generated by the other, they generate the same sigma-algebra. Also, if $\mathcal{A}\subseteq\mathcal{B}$, then $\sigma_X(\mathcal{A})\subseteq\sigma_X(\mathcal{B})$: the latter is a sigma-algebra containing $\mathcal{A}$, so the universal property applies.

Worked Example: The Sigma-Algebra Generated by One Set

Let $A\subseteq X$. If $A$ is neither empty nor all of $X$, the sigma-algebra generated by the single set $A$ is

\sigma_X(\{A\})=\{\varnothing,A,X\setminus A,X\}.

To verify this, denote the displayed family by $\mathcal{F}$. It contains $X$. Taking complements exchanges $\varnothing$ with $X$ and exchanges $A$ with $X\setminus A$, so $\mathcal{F}$ is closed under complements. Any countable union of members of $\mathcal{F}$ is the union of some selection from these four sets. If the selection contains $X$, its union is $X$. If it contains both $A$ and $X\setminus A$, its union is $X$. Otherwise, the union is one of $\varnothing,A,X\setminus A$. Thus $\mathcal{F}$ is a sigma-algebra containing $A$.

Now let $\mathcal{G}$ be any sigma-algebra containing $A$. It contains $X$, $\varnothing$, and $X\setminus A$, by the sigma-algebra properties. Hence $\mathcal{F}\subseteq\mathcal{G}$. The family $\mathcal{F}$ is therefore the smallest sigma-algebra containing $A$. If $A=\varnothing$ or $A=X$, the generated sigma-algebra is instead the trivial sigma-algebra $\{\varnothing,X\}$, which is already the family in the formula after repeated sets are removed.

Atoms from Finitely Many Generators

For finitely many generating sets, the generated sigma-algebra has a precise description. Each point of $X$ has a membership pattern: for every generator, the point either belongs to it or does not. Points with the same pattern cannot be separated by any set in the generated sigma-algebra. The nonempty groups of points with identical patterns are called the atoms associated with the generators.

Definition: Given $A_1,\ldots,A_n\subseteq X$, an atom associated with these sets is a nonempty set of the form $$ \bigcap_{i=1}^{n} C_i, \qquad \text{where each }C_i\text{ is either }A_i\text{ or }X\setminus A_i. $$ The choices of $C_i$ specify one membership pattern. Empty intersections of this form are discarded.

There are at most $2^n$ such patterns. Different nonempty atoms are disjoint: if two patterns differ at index $j$, one atom is contained in $A_j$ and the other in $X\setminus A_j$. Every point has exactly one membership pattern, so the atoms partition $X$.

Theorem (Finite Generators and Their Atoms): Let $A_1,\ldots,A_n\subseteq X$, and let $\mathcal{Q}$ be the collection of their nonempty atoms. Then $\sigma_X(\{A_1,\ldots,A_n\})$ consists exactly of all unions of subcollections of $\mathcal{Q}$, including the empty union.

Proof. By the Sigma-Algebra of Unions of Partition Blocks theorem, the family $\mathcal{F}_{\mathcal{Q}}$ of all unions of atoms is a sigma-algebra on $X$. Each generator $A_i$ is a union of atoms: it consists precisely of the atoms whose membership pattern selects $A_i$, rather than $X\setminus A_i$, at index $i$. Therefore $\mathcal{F}_{\mathcal{Q}}$ contains every generator. By the universal property, $\sigma_X(\{A_1,\ldots,A_n\})\subseteq\mathcal{F}_{\mathcal{Q}}$.

For the other inclusion, each atom is a finite intersection of generators or their complements. The generated sigma-algebra contains the generators, is closed under complements, and is closed under finite intersections, so it contains every atom. There are at most $2^n$ atoms, so each union of atoms is a finite union. The generated sigma-algebra is closed under finite unions, and consequently contains every member of $\mathcal{F}_{\mathcal{Q}}$. Thus $\mathcal{F}_{\mathcal{Q}}\subseteq\sigma_X(\{A_1,\ldots,A_n\})$, proving equality. $\square$

Worked Example: Two Generators on a Six-Point Space

Let $X=\{a,b,c,d,e,f\}$, $A=\{a,b,c\}$, and $B=\{c,d\}$. The four membership patterns give these atoms:

A\cap B=\{c\},\qquad A\cap(X\setminus B)=\{a,b\},

(X\setminus A)\cap B=\{d\},\qquad (X\setminus A)\cap(X\setminus B)=\{e,f\}.

These sets are disjoint and their union is $X$. The theorem says that every member of $\sigma_X(\{A,B\})$ is a union of some of these four atoms. For example, $A$ is the union $\{c\}\cup\{a,b\}=\{a,b,c\}$, and $B$ is the union $\{c\}\cup\{d\}=\{c,d\}$. The complement of $A$ is $\{d\}\cup\{e,f\}=\{d,e,f\}$, also a union of atoms.

There are $2^4=16$ unions because each of the four atoms can be included or excluded independently. The generated sigma-algebra is not the power set of $X$: for instance, $\{a\}$ is not a union of these atoms, since the atom $\{a,b\}$ must be taken as a whole. The generators distinguish the four atoms from one another, but they do not distinguish $a$ from $b$, or $e$ from $f$.

Adding a Generator and Replacing Generators

The atom description also explains when an additional set changes a generated sigma-algebra. If the added set is already a member of the existing sigma-algebra, it adds no new information. If it cuts an atom into smaller pieces, the resulting sigma-algebra can become larger.

Worked Example: An Added Set That Changes Nothing

Continue with $X,A,B$ from the preceding example. Let $C=\{a,b,d\}$. This is a union of atoms: $C=\{a,b\}\cup\{d\}$. Therefore $C\in\sigma_X(\{A,B\})$. Since the existing sigma-algebra contains $A$, $B$, and $C$, the universal property gives

\sigma_X(\{A,B,C\})\subseteq\sigma_X(\{A,B\}).

On the other hand, $\{A,B\}\subseteq\{A,B,C\}$, so monotonicity of generation gives the reverse inclusion. Hence the two sigma-algebras are equal. By contrast, if we add $D=\{a\}$, then $D$ splits the former atom $\{a,b\}$ into $\{a\}$ and $\{b\}$. The finite-atoms theorem shows that the new generated sigma-algebra contains $\{a\}$, which the old one did not. Thus this added generator makes the sigma-algebra strictly larger.

A useful form of the same reasoning is that a collection of generators can be replaced by any collection that generates the same sigma-algebra. In particular, once a set is known to belong to $\sigma_X(\mathcal{A})$, adding it to $\mathcal{A}$ does not enlarge the generated sigma-algebra. More generally, if $\mathcal{A}\subseteq\sigma_X(\mathcal{B})$, then $\sigma_X(\mathcal{A})\subseteq\sigma_X(\mathcal{B})$. These comparisons let us choose whichever description of a sigma-algebra is easiest to use.

Why Generation Is Useful

The word “smallest” is more than a label: it turns a construction problem into a pair of inclusion arguments. To prove that a candidate family is the generated sigma-algebra, one can show both that it is a sigma-algebra containing the generators and that every one of its members must lie in any sigma-algebra containing those generators. In finite settings, the atom theorem makes the candidate family explicit. In general settings, the universal property is often the more practical tool.

A common pitfall is to assume that listing the generators lists all the sets in the generated sigma-algebra. Even one generator usually forces its complement and the empty set and the whole space to be included. With several generators, intersections of generators and their complements produce the membership-pattern atoms, and unions of those atoms must also be included. Another pitfall is to assume that every added set enlarges the sigma-algebra: if it is already generated by the existing collection, it changes nothing.

For finitely many generators, no further distinctions are possible beyond the atoms: every set in the generated sigma-algebra is a union of whole atoms. This gives an efficient way to count its members as well. If there are $m$ nonempty atoms, then there are exactly $2^m$ unions, because each atom is either selected or not selected. The next tutorial will apply the idea of generation to a central family of subsets of the real line.

Check Your Understanding

Use the minimality property and the atom description to answer these questions.

If $\mathcal{F}$ is a sigma-algebra containing every member of $\mathcal{A}$, what inclusion must hold between $\sigma_X(\mathcal{A})$ and $\mathcal{F}$?
Why do two distinct atoms associated with finitely many generators have empty intersection?
For two generators $A$ and $B$, how many membership patterns are possible before empty atoms are discarded?
When does adjoining a set $C$ to a generator collection leave the generated sigma-algebra unchanged?
If there are $m$ nonempty atoms, why does their union structure give exactly $2^m$ members?

What You'll Learn