Measure theory MOC

σ-algebra

A σ-algebra Σ ⊆P(𝑋) on a set 𝑋 is a set of subsets of 𝑋 with the following properties measure

1. universal set: 𝑋 ∈Σ
2. compliment: 𝐴 ∈Σ ⟹ 𝑋 ∖𝐴 ∈Σ
3. countable union: {𝐴𝑖}∞𝑖=1 ⊆Σ ⟹ ⋃∞𝑖=1𝐴𝑖 ∈Σ

it immediately follows from De Morgan’s laws Σ is closed under

4. countable intersection: {𝐴𝑖}∞𝑖=1 ⊆Σ ⟹ ⋂∞𝑖=1𝐴𝑖 ∈Σ
5. set difference: 𝐴,𝐵 ∈Σ ⟹ 𝐴 ∖𝐵 ∈Σ

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