Probability Theory — Appendices
Appendix A — Measure Theory Needed for Probability A.1 Semirings and π-systems A semiring of sets is a primitive domain on which one can define a premeasure before extending it to a σ-algebra. A typical semiring is the family of half-open intervals ( 𝑎 , 𝑏 ] ( a , b ] in 𝑅 R , or rectangles of the form ( 𝑎 1 , 𝑏 1 ] × ⋯ × ( 𝑎 𝑑 , 𝑏 𝑑 ] ( a 1 , b 1 ] × ⋯ × ( a d , b d ] in 𝑅 𝑑 R d . Semirings are useful because complicated measurable sets are built from simpler geometric blocks, while measures are often first defined on those blocks. A π-system is a collection 𝑃 P of sets closed under finite intersections: 𝐴 , 𝐵 ∈ 𝑃 ⇒ 𝐴 ∩ 𝐵 ∈ 𝑃 . A , B ∈ P ⇒ A ∩ B ∈ P . Rectangles form a π-system. Cylinder sets in product spaces form a π-system. Half-lines ( − ∞ , 𝑡 ] ( − ∞ , t ] form a π-system generating the Borel σ-algebra on 𝑅 R . The value of π-systems is uniqueness. If two probability measures agree on a π-system 𝑃 P , and 𝑃 P generates 𝐹 F , then under stan...