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

Corollary to Inductive Construction of Sigma-Algebra Generated by Collection of Subsets
Let $\mathcal E$ be a set of sets which are subsets of some set $X$.

Let $\sigma\left({\mathcal E}\right)$ be the $\sigma$-algebra generated by $\mathcal E$.

Let the cardinality of $\mathcal E$ satisfy:


 * $\operatorname{card}\left({\N}\right) \le \operatorname{card}\left({\mathcal E}\right) \le \mathfrak c$

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

Then:


 * $\operatorname{card}\left({ \sigma\left({\mathcal E}\right) }\right) = \mathfrak c$

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


 * $\displaystyle \bigcup_{\alpha \mathop \in \Omega} \mathcal E_{\alpha} = \sigma\left({\mathcal E}\right)$

Thus by Leibniz's Law:


 * $\left \vert {\displaystyle \bigcup_{\alpha \mathop \in \Omega} \mathcal E_{\alpha} }\right \vert = \left \vert { \sigma\left({\mathcal E}\right) } \right \vert$

By the definition of union:


 * $\mathcal E_\alpha \subseteq \sigma\left({\mathcal E}\right)$ for all $\alpha \in \Omega$.

Thus:


 * $\left \vert { \mathcal E_\alpha }\right \vert \le \left \vert {\sigma\left({\mathcal E}\right)}\right \vert$ for all $\alpha \in \Omega$.

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


 * $\left \vert { \Omega } \right \vert \le \mathfrak c$

By Cardinality of Infinite Union of Infinite Sets:


 * $\left \vert {\displaystyle \bigcup_{\alpha \mathop \in \Omega} \mathcal E_{\alpha} }\right \vert \le \mathfrak c$

Thus:


 * $\left \vert { \sigma\left({\mathcal E}\right) } \right \vert \le \mathfrak c$

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


 * $\left \vert { \sigma\left({\mathcal E}\right) } \right \vert \ge \mathfrak c$

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