Improved Bonferroni Inequalities via Abstract Tubes - ReadingSample

1 Introduction and Overview Many problems in combinatorics, number theory, probability theory, reliability theory and statistics can be solved by applying a unifying method, which is known as the principle of inclusion-exclusion. The principle of inclusion-exclusion expresses the indicator function of a union of finitely many sets as an alternating sum of indicator functions of their intersections. More precisely, for any finite family of sets {A v } v∈V the principle of inclusion-exclusion states that χ v∈V A v = I⊆V I =∅ (−1) |I|−1 χ i∈I A i , (1.1) where χ(A) denotes the indicator function of A with respect to some universal set Ω, that is, χ(A)(ω) = 1 if ω ∈ A, and χ(A)(ω) = 0 if ω ∈ Ω\A. Equivalently, (1.1) can be expressed as χ v∈V ßA v = I⊆V (−1) |I| χ i∈I A i , where ßA v denotes the complement of A v in Ω and i∈∅ A i = Ω. A proof by induction on the number of sets is a common task in undergraduate courses. Usually, the A v 's are measurable with respect to some finite measure µ on a σ-field of subsets of Ω. Integration of the indicator function identity (1.1) with respect to µ then gives µ v∈V A v = I⊆V I =∅ (−1) |I|−1 µ i∈I A i , (1.2) which now expresses the measure of a union of finitely many sets as an alternating sum of measures of their intersections. The step leading from (1.1) to (1.2) is referred to as the method of indicators [GS96b]. Naturally, two special cases are of particular interest, namely the case where µ is the counting measure on the power set of Ω, and the case where µ is a probability measure on a σ-field of subsets of Ω. These special cases are sometimes attributed to Sylvester (1883) and Poincaré (1896), although the second edition of Montmort's book " Essai d'Analyse sur les Jeux de Hazard " , which appeared in 1714, already contains an implicit use of the method, based on a letter by N. Bernoulli in 1710. A first