1.4. Abstract measure spaces 75 Remark 1.4.20. The empty set axiom is needed in order to rule out the degenerate situation in which every set (including the empty set) has infinite measure. Example 1.4.21. Lebesgue measure m is a finitely additive measure on the Lebesgue σ-algebra, and hence on all sub-algebras (such as the null al- gebra, the Jordan algebra, or the elementary algebra). In particular, Jordan measure and elementary measure are finitely additive (adopting the con- vention that co-Jordan measurable sets have infinite Jordan measure, and co-elementary sets have infinite elementary measure). On the other hand, as we saw in previous notes, Lebesgue outer measure is not finitely additive on the discrete algebra, and Jordan outer measure is not finitely additive on the Lebesgue algebra. Example 1.4.22 (Dirac measure). Let x X and B be an arbitrary Boolean algebra on X. Then the Dirac measure δx at x, defined by set- ting δx(E) := 1E(x), is finitely additive. Example 1.4.23 (Zero measure). The zero measure 0: E 0 is a finitely additive measure on any Boolean algebra. Example 1.4.24 (Linear combinations of measures). If B is a Boolean algebra on X, and μ, ν : B [0, +∞] are finitely additive measures on B, then μ + ν : E μ(E) + ν(E) is also a finitely additive measure, as is cμ: E c × μ(E) for any c [0, +∞]. Thus, for instance, the sum of Lebesgue measure and a Dirac measure is also a finitely additive measure on the Lebesgue algebra (or on any of its sub-algebras). Example 1.4.25 (Restriction of a measure). If B is a Boolean algebra on X, μ: B [0, +∞] is a finitely additive measure, and Y is a B-measurable subset of X, then the restriction μ Y : B Y [0, +∞] of B to Y , defined by setting μ Y (E) := μ(E) whenever E B Y (i.e. if E B and E Y ), is also a finitely additive measure. Example 1.4.26 (Counting measure). If B is a Boolean algebra on X, then the function #: B [0, +∞] defined by setting #(E) to be the cardinality of E if E is finite, and #(E) := +∞ if E is infinite, is a finitely additive measure, known as counting measure. As with our definition of Boolean algebras and σ-algebras, we adopted a “minimalist” definition so that the axioms are easy to verify. But they imply several further useful properties: Exercise 1.4.20. Let μ: B [0, +∞] be a finitely additive measure on a Boolean σ-algebra B. Establish the following facts:
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