Sigma-Algebra as Magma of Sets

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The concept of $\sigma$-algebra is an instance of a magma of sets.


It will suffice to define partial mappings such that the axiom for a magma of sets crystallises into the axioms for a $\sigma$-algebra.

Let $X$ be any set, and let $\powerset X$ be its power set.


$\phi_1: \powerset X \to \powerset X: \map {\phi_1} S := X$
$\phi_2: \powerset X \to \powerset X: \map {\phi_2} S := X \setminus S$
$\phi_3: \powerset X^\N \to \powerset X: \map {\phi_3} {\sequence {S_n}_{n \mathop \in \N} } := \ds \bigcup_{n \mathop \in \N} S_n$

It is blatantly obvious that $\phi_1, \phi_2$ and $\phi_3$ capture the axioms for a $\sigma$-algebra.