We know that $\mathbb{X} = (0,1]$,and $\mathcal{M} = \{\emptyset,\text{finite unions of disjoint intervals that are open on the left and closed on the right} \}$ is not a $\sigma$-algebra.
However,if I let $\mathcal{M_1} = \{\emptyset,\text{countable unions of disjoint intervals that are open on the left and closed on the right} \}$, is $M_1$ a $\sigma$-algebra?
Best Answer
No it isn't.
The intervals $(1-\frac1n,1-\frac1{n+1}]$, $n=1,2,\dots$ are disjoint, the countable union is $(0,1)$, whose complement in $(0,1]$ is the singleton $\{1\}$ not in $\mathcal{M}_1$.