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Let $\mathcal M$ be an infinite $\sigma$-algebra.

I want to show that $\mathcal M$ contains an infinite subset of pairwise disjoint sets.

Although this question has been asked here before, I want to know if my particular approach is correct.

Let $\{E_n\}_{n=1}^\infty$ be an infinite subset of $\mathcal M$. Then $\{E_n\}_{n=1}^\infty$ is partially ordered with respect to set inclusion. Let $E_p,E_q\in\{E_n\}_{n=1}^\infty$ and say that $E_p\sim E_q$ if and only if $E_p$ and $E_q$ belong to the same chain. Since $\sim$ is an equivalence relation, let $\{\mathcal E_i\}_{i\in I}$ be the partition of $\{E_n\}_{n=1}^\infty$ into its equivalence classes.

If $I$ is finite, then there is a $j\in I$ such that $\mathcal E_j$ is infinite. Let $\{F_n\}_{n=1}^\infty$ be an infinite subset of $\mathcal E_j$ such that $F_n\subseteq F_{n+1}$ for all positive integers $n$, and let $\{G_n\}_{n=1}^\infty$ be such that $G_1=F_1$ and $G_n=F_n\setminus\bigcup_{k=1}^{n-1}F_k$. Then $\{G_n\}_{n=1}^\infty$ is an infinite subset of pairwise disjoint sets.

If $I$ is infinite, then for every $i\in I$, let $F_i\in\mathcal E_i$. Then $\{F_i\}_{i\in I}$ is an infinite subset of $\mathcal M$, and, as above, we can construct an infinite subset $\{G_n\}_{n=1}^\infty$ of pairwise disjoint sets.

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  • $\begingroup$ Nice proof! I like the way you incorporated ideals from abstract algebra into measure theory. $\endgroup$ – Landon Carter Oct 5 '16 at 2:34

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