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Unformatted text preview: 1. Prove Proposition 3.1. Proposition 3.1 Let be a signed measure on . If is an increasing sequence in , then lim . If is a decreasing sequence in and is finite, then lim . Answer) Let be a signed measure on . a. If is an increasing sequence in , if we set , then the sets are disjoint, so lim lim lim b. If is a decreasing sequence in , and , then let . Then , , and By (a) we have lim lim lim Subtracting from both sides gives lim 2. If is a signed measure, is -null iff . Also, if and are signed measure, iff iff and . Answer ) Let be a signed measure. If is -null, and if is a Hahn decomposition of , then and ; hence, . Conversely, if , then . Thus if , then and , from which it follows that . As this is true for any measurable , is -null. 3. Let be a signed measure on . a. . b. If , ....
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- Fall '08
- measure, Yonsei University, Signed measure, Complex measure, KC Jung