Lebesgue Integrals

Exercise 11: Answer 15-18

Exercises 11 Math 250 October 7, 2020 Given 1. a measure space ([2,.F, P) and 2. a simple, non-negative, measurable function X on {2. Show that 3. the function : .7: > [0,00] E v> / X dP E is a measure on (QT). That is, that 4. 13 is countably additive, and 5. 15 is not identically 00. First, verify that 6. is not identically 00. To establish the countable additivity of , let 7. A1, A2, . . . be a sequence of pairwise disjoint members of .7: and 8. A 2 U2; An. We Wish to show that 9. 13M) = 22:13am. If 10. :51, $2, . . . ,:cm are all the distinct values of X, and 11. B,- = X'1({$i}), i = 1,2,...,m, then 12. B1,B2,...,Bm are pairwise disjoint, and 13. X : 2::1 331-131.. Verify that 14. every mi is nonnegative, 15. every Bi is a member of F, 16. for every E E .73, 13(E) : 21:1 xiP(Bi m E), and 17. for every 2', P(Bi 0 A) 2 22:1 P(Bz 0 An), and conclude that 18. HA) : 2;: 115m")