i was wondering whether it makes any difference (or whether it is even true) that an uncountably infinite union/intersection of sets that are elements of a sigma algebra is again an element of the sigma algebra?
2026-05-13 17:27:37.1778693257
Uncountably infinite union/intersection in sigma-algebra
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If $X$ is a set and $\mathcal B\subset 2^X$ a $\sigma$-algebra on $X$, it's not true. For example, take $\mathbb R$ endowed with the $\sigma$-algebra $\mathcal B$ of the subset of $\mathbb R$ which are at most countable, and their complements. Then write $(0,\infty):=\bigcup_{x\gt 0}\{x\}$.