I have already proved several similar questions but for the Borel sigma algebra on the reals and I am now a bit stuck when working with the extended reals.
2026-03-30 11:04:53.1774868693
Trying to prove $\{[−\infty, a)\}_{a\in\mathbb{R}}$ generates The borel sigma algebra of the extended reals. Any hints?
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Taking complements, we see each $[a,\infty]$ is obtained. Since $(a,\infty]=\cup [a+1/n,\infty],$ we obtain each $(a,\infty].$ Given reals $a<b,$ we obtain $(a,\infty]\cap [-\infty,b) =(a,b).$ That's a good start ...