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Unformatted text preview: the Borel σ -algebra
of [0, 1], is the σ -algebra generated by the collection of open subsets of [0, 1]. As the next
exercise shows, we can equivalently think of B1 as the restriction of B to [0, 1]. Note that
B1 ⊂ B as collections of sets, but not as σ -algebras. That is, B1 is not a sub-σ -algebra of B .
The reason, of course, is that B is a σ -algebra of subsets of R whereas B1 is a σ -algebra of
subsets of [0, 1]; in order for one σ -algebra to be a sub-σ -algebra of another σ -algebra, it is
necessarily the case that the underlying samp...
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