205A Homework #2, due Tuesday 11 September. 1. [similar Bill. 2.15] Let B be the Borel subsets of R. For B 2 B dene (B) = 1 if (0; ) B for some > 0 = 0 if not (a) Show that is not nitely additive on B. (b) Show that is nitely additive but not countably additive on the eld B0 of nite disjoint unions of intervals (a; b]. 2. Show that, in the denition of a probability measure on a measurable space (S; S), we may replace countably additive by itely additive, and satises if An # then (An) ! 0 : 3. [similar Durr. A.2. Document Preview: 205A Homework #2, due Tuesday 11 September. 1. [similar Bill. 2.15] Let B be the Borel subsets of R. For B 2 B dene (B) = 1 if (0; ) B for some > 0 = 0 if not (a) Show that is not nitely additive on B. (b) Show that is nitely additive but not countably additive on the eld B0 of nite disjoint unions of intervals (a; b]. 2. Show that, in the denition of a probability measure on a measurable space (S; S), we may replace countably additive by itely additive, and satises if An # then (An) ! 0 : 3. [similar Durr. A.2.1] Give an example of a measurable space (S; S), a collection A and probability measures and such that (i) (A) = (A) for all A 2 A (ii) S = (A) (iii) 6= . Note: this can be done with S = f1; 2; 3; 4g 4. [similar Durr. Lemma A.2.1] Let be a probability measure on (S; S), where S = (F) for a eld F. Show that for each B 2 S and > 0 there exists A 2 F such that (BA) 0. ShR ow that there exists a continuous function f : [0; 1] ! R such that jf(x) ?? g(x)j dx . 2 Attachments: hw-Part2.pdf; 205A Homework #2, due Tuesday 11 September. 1. [similar Bill. 2.15] Let B be the Borel subsets of R. For B 2 B dene (B) = 1 if (0; ) B for some > 0 = 0 if not (a) Show that is not nitely additive on B. (b) Show that is nitely additive but not countably additive on the eld B0 of nite disjoint unions of intervals (a; b]. 2. Show that, in the denition of a probability measure on a measurable space (S; S), we may replace countably additive by itely additive, and satises if An # then (An) ! 0 : 3. [similar Durr. A.2. Document Preview: 205A Homework #2, due Tuesday 11 September. 1. [similar Bill. 2.15] Let B be the Borel subsets of R. For B 2 B dene (B) = 1 if (0; ) B for some > 0 = 0 if not (a) Show that is not nitely additive on B. (b) Show that is nitely additive but not countably additive on the eld B0 of nite disjoint unions of intervals (a; b]. 2. Show that, in the denition of a probability measure on a measurable space (S; S), we may replace countably additive by itely additive, and satises if An # then (An) ! 0 : 3. [similar Durr. A.2.1] Give an example of a measurable space (S; S), a collection A and probability measures and such that (i) (A) = (A) for all A 2 A (ii) S = (A) (iii) 6= . Note: this can be done with S = f1; 2; 3; 4g 4. [similar Durr. Lemma A.2.1] Let be a probability measure on (S; S), where S = (F) for a eld F. Show that for each B 2 S and > 0 there exists A 2 F such that (BA) 0. ShR ow that there exists a continuous function f : [0; 1] ! R such that jf(x) ?? g(x)j dx . 2 Attachments: hw-Part2.pdf
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