MAT 324: Measure Theory
Stony Brook University
Alexander Davison
2025-09-02
Contents
Measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
𝜎-algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Smallest containing 𝜎-algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
Borel 𝜎-algebra on ℝ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
Lebesgue 𝜎-algebra on ℝ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
Measure
𝜎-algebra
Recall that for open intervals 𝐼 = (𝑎, 𝑏) ≠ ∅, the length ℓ(𝐼) = 𝑏 − 𝑎. Also recall:
Definition. The outer measure of 𝐴 ⊂ ℝ, denoted as |𝐴|, is defined by
∞
∞
|𝐴| ≔ inf{∑ ℓ(𝐼𝑘 ) : 𝐴 ⊂ ⋃ 𝐼𝑘 ; and 𝐼𝑘 ≠ ∅ open interval ∀𝑘}.
𝑘=0
𝑘=0
While the outer measure is a powerful concept for measurement on ℝ, it has a notable limitation:
Proposition. ∃𝐴, 𝐵 ⊂ ℝ where 𝐴 ∩ 𝐵 = ∅ and |𝐴 + 𝐵| ≠ |𝐴| + |𝐵|.
To address this subadditivity problem, we may instead consider only the collection of “good” subsets 𝒮. We
would like to have ∅ ∈ 𝒮, closure under complements, and closure under countable unions.
Definition. A 𝜎-algebra is a collection of subsets of a set 𝑋 where
(0) ∅ ∈ 𝒮;
(1) 𝐴 ∈ 𝒮 ⟹ 𝑋 − 𝐴 ∈ 𝒮;
∞
(2) 𝐴0 , 𝐴1 , … ∈ 𝒮 ⟹ ⋃ 𝐴𝑘 ∈ 𝒮.
𝑘=0
We may equivalently substitute (0′) for (0), (1′) for (1), or (2′) for (2) where
(0') 𝑋 ∈ 𝒮;
(1') 𝐴, 𝐵 ∈ 𝒮 ⟹ 𝐴 − 𝐵 ∈ 𝒮;
∞
(2') 𝐴0 , 𝐴1 , … ∈ 𝒮 ⟹ ⋂ 𝐴𝑘 ∈ 𝒮.
𝑘=0
Some examples of 𝜎-algebras on 𝑋 include the boring {∅, 𝑋} and 2𝑋 , and the more interesting
{𝐴 ⊂ 𝑋 : 𝐴 countable or 𝑋 − 𝐴 countable}.
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MAT 324: Measure Theory
Stony Brook University
Alexander Davison
2025-09-02
Smallest containing 𝜎-algebra
Lemma. Suppose {𝒮𝛼 }𝛼∈𝒜 is a collection of 𝜎-algebras on 𝑋. Then ⋂𝛼∈𝒜 𝒮𝛼 is also a 𝜎-algebra.
Corollary. Suppose 𝒮0 ⊂ 2𝑋 . Then
𝜎(𝒮0 ) ≔ ⋂{𝑆0 ⊂ 𝑆 : 𝑆 𝜎-algebra}
is the smallest 𝜎-algebra containing 𝒮0 , also called the 𝜎-algebra generated by 𝑆0 .
Borel 𝜎-algebra on ℝ
Definition. The Borel 𝜎-algebra on ℝ, denoted as ℬ, is defined by
ℬ ≔ 𝜎({𝒰 ⊂ ℝ open}).
Elements of the Borel 𝜎-algebra are called Borel sets.
Lebesgue 𝜎-algebra on ℝ
Definition. The Lebesgue 𝜎-algebra on ℝ, denoted as ℳ, is defined by
ℳ = {𝐴 ⊂ ℝ : ∀𝜀 ∈ ℝ>0 , ∃𝐴𝜀 closed where 𝐴𝜀 ⊂ 𝐴 and |𝐴 − 𝐴𝜀 | < 𝜀}.
Proposition. 𝐴 ⊂ ℝ closed ⟹ 𝐴 ∈ ℳ.
Proof. Let 𝐴𝜀 = 𝐴. Then |𝐴 − 𝐴𝜀 | = 0 < 𝜀. 证毕
Proposition. 𝐴 ⊂ ℝ where |𝐴| = 0 ⟹ 𝐴 ∈ ℳ.
Proof. Let 𝐴𝜀 = ∅. Then |𝐴 − 𝐴𝜀 | = |𝐴| = 0 < 𝜀. 证毕
Proposition. ℳ is a 𝜎-algebra on ℝ.
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