1.2.2: Let u be a measure on an algebra F, and let A, B, A₁,..., Ak EF, 1 ≤ k <∞o. Then, (i) (Monotonicity) μ(A) ≤μ(B) if AC B; (ii) (Finite subadditivity) (A₁ U...UA) ≤ (A₁)+...+μ(A); (iii) (Inclusion-exclusion formula) If μ(A₁) <∞o for all i = 1,..., k, then k μ(4U...UA») = Σμ(4) - Σ. MADA) i=1 1

Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter2: Equations And Inequalities
Section2.4: Complex Numbers
Problem 61E
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Α, Β, Α1,...,.
1.2.2: Let u be a measure on an algebra F, and let
..,Ak €F,1<k < m. Then,
(i) (Monotonicity) µ(A) ≤ µ(B) if AC B;
(ii) (Finite subadditivity) μ(A1 U... U Ak) < μ(A1) + ... + μ(Ak);
(iii) (Inclusion-exclusion formula) If μ(A;) < oo for all i = 1,...,k, then
-
μ(AU... Aκ) = Σμ(4) - ΣΜ(ΑΠΑ)
i=1
1<i<j<k
+...+(-1)k−1μ(Α1 Π... Π.Α.).
Transcribed Image Text:Α, Β, Α1,...,. 1.2.2: Let u be a measure on an algebra F, and let ..,Ak €F,1<k < m. Then, (i) (Monotonicity) µ(A) ≤ µ(B) if AC B; (ii) (Finite subadditivity) μ(A1 U... U Ak) < μ(A1) + ... + μ(Ak); (iii) (Inclusion-exclusion formula) If μ(A;) < oo for all i = 1,...,k, then - μ(AU... Aκ) = Σμ(4) - ΣΜ(ΑΠΑ) i=1 1<i<j<k +...+(-1)k−1μ(Α1 Π... Π.Α.).
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