1.4. Abstract measure spaces 77 Example 1.4.28. Lebesgue measure is a countably additive measure on the Lebesgue σ-algebra, and hence on every sub-σ-algebra (such as the Borel σ- algebra). Example 1.4.29. The Dirac measures from Exercise 1.4.22 are countably additive, as is counting measure. Example 1.4.30. Any restriction of a countably additive measure to a measurable subspace is again countably additive. Exercise 1.4.22 (Countable combinations of measures). Let (X, B) be a measurable space. (i) If μ is a countably additive measure on B, and c ∈ [0, +∞], then cμ is also countably additive. (ii) If μ1,μ2,... are a sequence of countably additive measures on B, then the sum ∑∞ n=1 μn : E → ∑∞ n=1 μn(E) is also a countably additive measure. Note that countable additivity measures are necessarily finitely additive (by padding out a finite union into a countable union using the empty set), and so countably additive measures inherit all the properties of finitely ad- ditive properties, such as monotonicity and finite subadditivity. But one also has additional properties: Exercise 1.4.23. Let (X, B,μ) be a measure space. (i) (Countable subadditivity) If E1,E2,... are B-measurable, then we have μ( ∞ n=1 En) ≤ ∑∞ n=1 μ(En). (ii) (Upwards monotone convergence) If E1 ⊂E2 ⊂... are B-measurable, then μ( ∞ n=1 En) = lim n→∞ μ(En) = sup n μ(En). (iii) (Downwards monotone convergence) If E1 ⊃ E2 ⊃ . . . are B- measurable, and μ(En) ∞ for at least one n, then μ( ∞ n=1 En) = lim n→∞ μ(En) = inf n μ(En). Show that the downward monotone convergence claim can fail if the hy- pothesis that μ(En) ∞ for at least one n is dropped. (Hint: Mimic the solution to Exercise 1.2.11.) Exercise 1.4.24 (Dominated convergence for sets). Let (X, B,μ) be a mea- sure space. Let E1,E2,... be a sequence of B-measurable sets that converge to another set E, in the sense that 1En converges pointwise to 1E.

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