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Unformatted text preview: Mathematics 105, Spring 2004 M. Christ Midterm Exam #2 Comments Distribution of scores : There were 50 points possible. The highest score was 50 and the third highest was 39. The median was 25 1 2 , the 75th percentile was 35, and the 25th percentile score was 21. Final course grades will be based 1 15% on MT1, 20% on this exam, 20% on problem sets, and 45% on the final exam. (1a) In defining a Lebesgue measurable set, a few students wrote | G \ A | < instead of | G \ A | e < . The distinction is that | G \ A | is not defined unless one already knows that G \ A is measurable. In testing measurability, one uses exterior measure, which is defined for all sets. (1b) If f is measurable then R E f d is defined if and only if the two numbers R E f + d and R E f- d are not both equal to + . A few students also mentioned the case where both integrals equal- . But this can never arise since f + , f- are by definition nonnegative functions. The question did not call for a definition of R E f d , so there was no need for you to reproduce the chain of definitions. (1c) A few people forgot to put absolute value signs around det ( T ). Clearly these are needed; the Lebesgue measure of T ( A ) cant be negative! (2a) The relevant example is the fat or generalized Cantor set, which we discussed in a homework problem. Given (0 , 1), this set is constructed by deleting from [0 , 1] a subinterval of length / 3, leaving two subintervals of equal lengths. From the middle of each of those we delete a subinterval of length / 3 2 , leaving four subintervals of equal lengths. From the middle of each of those we delete a subinterval of length / 3 3 , and so forth; the fat Cantor set is the set of all points which are never deleted if this process is continued through infinitely many steps....
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This note was uploaded on 05/07/2008 for the course MATH 105 taught by Professor Michaelchrist during the Spring '04 term at University of California, Berkeley.
- Spring '04