Sigma-Algebra as Magma of Sets

Theorem

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

Proof

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.

Define:

$\phi_1: \powerset X \to \powerset X: \map {\phi_1} S := X$

$\phi_2: \powerset X \to \powerset X: \map {\phi_2} S := X \setminus S$

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

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

This needs considerable tedious hard slog to complete it. In particular: Bladibladiblabla, i.e. show that they are mappings, and that the MoS property translates into the $\sigma$-axiom To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Finish}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

$\blacksquare$