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Theorem 5.3. The Borel σ -algebra B is generated by intervals of the form (−∞, a] where
a ∈ Q is a rational number.
Proof. Let O0 denote the collection of all open intervals. Since every open set in R is an
at most countable union of open intervals, we must have σ (O0 ) = B . Let D denote the
collection of all intervals of the form (−∞, a], a ∈ Q. Let (a, b) ∈ O0 for some b > a with a,
b ∈ Q. Let
1
an = a +
n
so that an ↓ a as n → ∞, and let
1
bn = b −
n
so that bn ↑ b as n → ∞. Thus,
(a, b) = ∞
n=1 (an , bn ] = ∞
n=1 {(−∞, bn ] ∩ (−∞, an ]c } which implies that (a, b) ∈ σ (D). That is, O0 ⊆ σ (D) so that σ (O0 ) ⊆ σ (D). However,
every element of D is a closed set which implies that
σ (D ) ⊆ B .
5–2 This gives the chain of containments
B = σ ( O0 ) ⊆ σ ( D ) ⊆ B
and so σ (D) = B proving the theorem. The Borel sets of [0, 1]
If we now consider the set [0, 1] ⊂ R as the sample space, then B1 ,...

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