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**Unformatted text preview: **open set, then, by deﬁnition,
Ac is closed and so Ac ∈ O.) However, we know that σ (O), the σ -algebra generated by O,
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exists and satisﬁes O σ (O) ⊆ 2R . This leads to the following deﬁnition.
Deﬁnition. The Borel σ -algebra of R, written B , is the σ -algebra generated by the open
sets. That is, if O denotes the collection of all open subsets of R, then B = σ (O).
Since B is a σ -algebra, we see that it necessarily contains all open sets, all closed sets,
all unions of open sets, all unions of closed sets, all intersections of closed sets, and all
intersections of open sets.
Exercise 5.1. The purpose of this exercise is to is to remind you of some facts about open
and closed sets. Suppose that {E1 , E2 , . . .} is an arbitrary collection of open subsets of R,
and suppose that {F1 , F2 , . . .} is an arbitrary collection of closed subsets of R. Prove that
(a)
(b)
(c)
(d) ∞ j =1 ∞ j =1 ∞ j =1 ∞ j =1 Ej is necessarily open,
Ej need not be open,
Fj is necessarily closed, and
Fj need not be closed. We often call a countable intersection of open sets a Gδ set (from the German Gebeit for
open and Durchschnitt for interse...

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