| Problem Set |
Chapter |
Problems |
|---|---|---|
| 27 | ∗43 | In Note L.26 we pointed out that a linear combination f of characteristic functions is measurable if and only if each of the characteristic functions is for a measurable set. Prove this statement. (The term linear combination is being used here just as it is used in linear algebra (MATH 255/256 or MATH 352) or differential equations (MATH 231/333). For the definition of this term, given in the context of vectors xi rather than functions, see Rudin, p. 204.) |
| 11 | 3 -- In proving this result, you may make the additional assumption that for each x in X, (fn(x)) is a bounded sequence. (That is, there is a number Mx > 0 for which |fn (x)| ≤ Mx. As x changes, we must assume Mx does to.) Challenge: Prove the result without this additional assumption (i.e., the result as it is stated in Rudin.) |
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| n | manditory problem |
| ∗n | manditory problem, not from text |
| (n) | helper problem |
| [n] | ungraded problem |
| [∗n] | ungraded problem, not from text |
| {n} | optional problem |