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Show that the σ-algebras generated by the collection of all intervals of the form [a,b]⊂R and by the collection of all the intervals of the form (−∞,b]⊂R are equivalent.


i am having trouble with this one

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For one direction, each interval of the form $(-\infty,b]$ could be represented as the countable union of intervals of the other type, namely $(-\infty,b]=\cup_{n=0}^\infty [b-n-1,b-n]$.

For the other direction, each $[a,b]$ equals $(-\infty,b]\setminus(-\infty,a)$, and $(-\infty,a)$ in turn equals $\cup_{n=1}^\infty (-\infty,a-\frac1n]$. That is, $[a,b]=(-\infty,b]\setminus\bigl(\cup_{n=1}^\infty (-\infty,a-\frac1n]\bigr)$.

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