76 1. Measure theory (i) (Monotonicity) If E, F are B-measurable and E F , then μ(E) μ(F ). (ii) (Finite additivity) If k is a natural number, and E1,...,Ek are B- measurable and disjoint, then μ(E1∪...∪Ek) = μ(E1)+...+μ(Ek). (iii) (Finite subadditivity) If k is a natural number, and E1,...,Ek are B-measurable, then μ(E1 . . . Ek) μ(E1) + . . . + μ(Ek). (iv) (Inclusion-exclusion for two sets) If E, F are B-measurable, then μ(E F ) + μ(E F ) = μ(E) + μ(F ). (Caution: Remember that the cancellation law a + c = b + c =⇒ a = b does not hold in [0, +∞] if c is infinite, and so the use of cancellation (or subtraction) should be avoided if possible.) One can characterise measures completely for any finite algebra: Exercise 1.4.21. Let B be a finite Boolean algebra, generated by a finite family A1,...,Ak of non-empty atoms. Show that for every finitely additive measure μ on B there exists c1,...,ck [0, +∞] such that μ(E) = 1≤j≤k:Aj⊂E cj. Equivalently, if xj is a point in Aj for each 1 j k, then μ = k j=1 cjδxj . Furthermore, show that the c1,...,ck are uniquely determined by μ. This is about the limit of what one can say about finitely additive mea- sures at this level of generality. We now specialise to the countably additive measures on σ-algebras. Definition 1.4.27 (Countably additive measure). Let (X, B) be a measur- able space. An (unsigned) countably additive measure μ on B, or measure for short, is a map μ: B [0, +∞] that obeys the following axioms: (i) (Empty set) μ(∅) = 0. (ii) (Countable additivity) Whenever E1,E2,... B are a countable se- quence of disjoint measurable sets, then μ( n=1 En)= ∑∞ n=1 μ(En). A triplet (X, B,μ), where (X, B) is a measurable space and μ: B [0, +∞] is a countably additive measure, is known as a measure space. Note the distinction between a measure space and a measurable space. The latter has the capability to be equipped with a measure, but the former is actually equipped with a measure.
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