Inductive Construction of Sigma-Algebra Generated by Collection of Subsets/Corollary
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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 : $\S 1.23$