Exercise 1.4.10 (Atomic algebras $\sigma$-algebras)

Question

Show that all atomic algebras are $\sigma$-algebras.

Elu Thingol · Accepted Answer

Let $\mathcal{A}$ be an atomic algera generated by the atoms $(A_\alpha)_{\alpha\in I}$. Let $\left(E\right)_{n\in\mathbb{N}}$ be the sequence of sets in $\mathcal{A}$. By definition, for each $i\in\mathbb{N}$ the set $E_i\in\mathcal{A}$ can be represented as a union of $J_i \subseteq I$-atoms $(A_\alpha)_{\alpha \in J_i}$. In other words,
$$\bigcup_{i=1}^{\infty} E_i = \bigcup_{i=1}^{\infty} \bigcup_{\alpha \in J_i} A_\alpha = \bigcup_{\alpha \in \bigcup_{i=1}^{\infty}J_i} A_\alpha.$$
But this implies that the union $\bigcup_{i=1}^{\infty} E_i$ is itself formed by atoms and thus $\bigcup_{i=1}^{\infty} E_i \in \mathcal{A}$ as desired.

Exercise 1.4.10 (Atomic algebras $\sigma$-algebras)

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Exercise 1.4.10 (Atomic algebras $\sigma$-algebras)

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