1.7. Outer measures, pre-measures, product measures 153 1.7.2. Pre-measures. In previous notes, we saw that finitely additive measures, such as elementary measure or Jordan measure, could be extended to a countably additive measure, namely Lebesgue measure. It is natural to ask whether this property is true in general. In other words, given a finitely additive measure μ0 : B0 [0, +∞] on a Boolean algebra B0, is it possible to find a σ-algebra B refining B0, and a countably additive measure μ: B [0, +∞] that extends μ0? There is an obvious necessary condition in order for μ0 to have a count- ably additive extension, namely that μ0 already has to be countably additive within B0. More precisely, suppose that E1,E2,E3,... B0 were disjoint sets such that their union n=1 En was also in B0. (Note that this latter property is not automatic as B0 is merely a Boolean algebra rather than a σ-algebra.) Then, in order for μ0 to be extendible to a countably additive measure, it is clearly necessary that μ0( n=1 En) = n=1 μ0(En). Using the Carath´ eodory extension theorem, we can show that this nec- essary condition is also sufficient. More precisely, we have Definition 1.7.7 (Pre-measure). A pre-measure on a Boolean algebra B0 is a finitely additive measure μ0 : B0 [0, +∞] with the property that μ0( n=1 En) = ∑∞ n=1 μ0(En) whenever E1,E2,E3,... B0 are disjoint sets such that n=1 En is in B0. Exercise 1.7.4. (i) Show that the requirement that μ0 is finitely additive can be relaxed to the condition that μ0(∅) = 0 without affecting the definition of a pre-measure. (ii) Show that the condition μ0( n=1 En) = ∑∞ n=1 μ0(En) can be re- laxed to μ0( n=1 En) n=1 μ0(En) without affecting the defini- tion of a pre-measure. (iii) On the other hand, give an example to show that if one performs both of the above two relaxations at once, one starts admitting objects μ0 that are not pre-measures. Exercise 1.7.5. Without using the theory of Lebesgue measure, show that elementary measure (on the elementary Boolean algebra) is a pre-measure. (Hint: Use Lemma 1.2.6. Note that one has to also deal with co-elementary sets as well as elementary sets in the elementary Boolean algebra.) Exercise 1.7.6. Construct a finitely additive measure μ0 : B0 [0, +∞] that is not a pre-measure. (Hint: Take X to be the natural numbers, take
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