Inductive Construction of Sigma-Algebra Generated by Collection of Subsets/Corollary

Corollary to Inductive Construction of Sigma-Algebra Generated by Collection of Subsets

Let $\EE$ be a set of sets which are subsets of some set $X$.

Let $\map \sigma \EE$ be the $\sigma$-algebra generated by $\EE$.

Let the cardinality of $\EE$ satisfy:

$\size \N \le \size \EE \le \mathfrak c$

where $\mathfrak c$ denotes the cardinality of the continuum.

Then:

$\size {\map \sigma \EE} = \mathfrak c$

Proof

By Inductive Construction of Sigma-Algebra Generated by Collection of Subsets:

$\ds \bigcup_{\alpha \mathop \in \Omega} \EE_{\alpha} = \map \sigma \EE$

Thus by Leibniz's Law:

$\size {\ds \bigcup_{\alpha \mathop \in \Omega} \EE_{\alpha} } = \size {\map \sigma \EE}$

By the definition of union:

$\EE_\alpha \subseteq \map \sigma \EE$ for all $\alpha \in \Omega$.

Thus:

$\size {\EE_\alpha} \le \size {\map \sigma \EE}$ for all $\alpha \in \Omega$.

By Corollary of Existence of Minimal Uncountable Well-Ordered Set:

$\size \Omega \le \mathfrak c$

By Cardinality of Infinite Union of Infinite Sets:

$\ds \size {\bigcup_{\alpha \mathop \in \Omega} \EE_{\alpha} } \le \mathfrak c$

Thus:

$\size {\map \sigma \EE} \le \mathfrak c$

By Cardinality of Infinite Sigma-Algebra is at Least Cardinality of Continuum:

$\size {\map \sigma \EE} \ge \mathfrak c$

The result follows from the Cantor-Bernstein-Schröder Theorem.

$\blacksquare$

Sources

• 1984: Gerald B. Folland: Real Analysis: Modern Techniques and their Applications : $\S 1.23$