Generated Sigma-Algebra Preserves Finiteness

Theorem
Let $X$ be a set, and let $A_1, \ldots, A_n \subseteq X$ be subsets of $X$.

Then $\map \sigma {\set {A_1, \ldots, A_n} }$ is a finite set, where $\sigma$ denotes generated $\sigma$-algebra.

Proof
Proceed by induction on $n$, that is, on the number of generators.

Basis for the Induction
The case $n = 0$ is verified as follows:


 * $\map \sigma \O = \set {\O, X}$

from Sigma-Algebra Generated by Empty Set.

Thus $\map \sigma \O$ is finite.

This is the basis for the induction.

Induction Hypothesis
Fix $n \in \N$.

Assume that $\map \sigma {\set {A_1, \ldots, A_n} }$ is finite.

This is the induction hypothesis.

Induction Step
This is the induction step:

For brevity, write $\Sigma_n$ for $\map \sigma {A_1, \ldots, A_n}$.

Observe that:


 * $\set {A_1, \ldots, A_{n + 1} } \subseteq \Sigma_n \cup \set {A_{n + 1} } \subseteq \Sigma_{n + 1}$

Hence Condition on Equality of Generated Sigma-Algebras applies to yield:


 * $\Sigma_{n + 1} = \map \sigma {\Sigma_n \cup \set {A_{n + 1} } }$

From Sigma-Algebra Extended by Single Set, it follows that:


 * $\map \sigma {\Sigma_n \cup \set {A_{n + 1} } } = \set {\paren {E_1 \cap A_{n + 1} } \cup \paren {E_2 \cap A_{n + 1}^c}: E_1, E_2 \in \Sigma_n}$

Thus there is a surjection $f: \Sigma_n \times \Sigma_n \to \Sigma_{n + 1}$, defined by:


 * $\map f {E_1, E_2} := \paren {E_1 \cap A_{n + 1} } \cup \paren {E_2 \cap A_{n + 1}^c}$

By Cardinality of Surjection:


 * $\card {\Sigma_{n + 1} } \le \card {\Sigma_n \times \Sigma_n}$

and the latter equals $\card {\Sigma_n}^2$ by Cardinality of Cartesian Product of Finite Sets.

By the induction hypothesis, this is necessarily finite.

Therefore, $\Sigma_{n + 1}$ is also finite.

The result follows by the Principle of Mathematical Induction.