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.

Let $I$ be an index set.

For $S \in \powerset X$, 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.

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Let $I$ be the closed (real) interval $[1, 3]$.

For every $i \in I = [1, 3]$, let $J_i = \N$ be the index set, and let:

$\phi_i: \powerset X^\N \to \powerset X$

be a partial mapping.

We have:

$X \in \SS$

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Also:

$X \setminus S \in \SS$

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From Union of Subsets is Subset/Subset of Power Set, we have:

$\ds \bigcup_{n \mathop \in \N} S_n \in \SS$

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It follows that for every indexed family $\family {S_n}_{n \mathop \in \N} \in \SS^{\N}$ in the domain of $\phi_i$ for every $i \in I$:

$\forall i \in [1, 3]: \map {\phi_i} {\family {S_n}_{n \mathop \in \N} } \in \SS$

It follows that $\SS$ is closed under $\phi_i$ for all $i \in I$.

So $\SS$ is a magma of sets for $\set {\phi_i: i \in I}$ on $X$

Hence the result.

$\blacksquare$