Problem Set
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Chapter
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Problems
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| 26 |
∗40 |
Show that if E is a measurable subset of S (as defined in Thm. L.24), then m(E) = 0. Here m denotes Lebesgue measure (Lebesgue outer measure arising from the set function m defined in 11.4, restricted to the σ-algebra of Thm. 11.10). (Hint: Define sets Ei = E + qi, where the qi are as in the proof of Thm. L.24.)
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∗41 |
Show that if A is any subset of (0, 1) for which the Lebesgue outer measure of A is positive, then A contains a nonmeasurable subset. (Hint: Define sets Ai to be the intersection of A and Si, where Si is as defined in Thm. L.24.)
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∗42 |
Prove directly (without appealing to Note (ii) of 11.14) that constant functions defined on X are measurable (this regardless of what σ-algebra is used to turn X into a measurable space).
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