Category:Generated Sigma-Algebra by Generated Monotone Class

This category contains pages concerning Generated Sigma-Algebra by Generated Monotone Class:

Let $X$ be a set, and let $\GG \subseteq \powerset X$ be a nonempty collection of subsets of $X$.

Suppose that $\GG$ satisfies the following condition:

$(1):\quad A \in \GG \implies \relcomp X A \in \GG$

that is, $\GG$ is closed under complement in $X$.

Then:

$\map {\mathfrak m} \GG = \map \sigma \GG$

where $\mathfrak m$ denotes generated monotone class, and $\sigma$ denotes generated $\sigma$-algebra.