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Here is exercise 1.1(c) from Folland's Real Analysis:

If $\mathcal{R}$ is a $\sigma$-ring, then $\mathcal{M}= \{ E\subset X : E \in \mathcal{R} \text{ or } E^c \in \mathcal{R} \} $ is a $\sigma$-algebra.

Recall that a family of sets $\mathcal{R} \subseteq \mathcal{P}(X)$ is a $\sigma$-ring if it is closed under differences (i.e., if $E,F \in \mathcal{R}$ then $E \setminus F \in \mathcal{R}$) and countable unions.

If I take $X = \mathbb{R}$, $\mathcal{R} = \{\{0\}\}$, which is definitely a $\sigma$-ring, then $\{0\}, \mathbb{R}\setminus \{0\} \in \mathcal{M}$, but $\mathbb{R} = \{0\} \cup \mathbb{R}\setminus \{0\} \not\in \mathcal{R}$, nor $\emptyset \in \mathcal{R}$. What's wrong here?

2 Answers 2