Exercises 11 Math 250 October 7, 2020 Given 1

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 non—negative, 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").

Exercises 14 Math 250 October 13, 2020 Given 1. a measure space (9, f, P) and 2. a non—negative7 measurable function X on .Q. Show that 3. the function 15 : .7: —> [0, 00] E r—> / X dP E is a measure on (97.73). That is, show that 4. I3 is countably additive, and 5. IE5 is not identically 00. First, verify that 6. 13 is not identically 00. To establish the countable additivity of P, let 7. 141,142, . . . be a sequence of pairwise disjoint members of J: and 8. B = U21 Ai. We wish to show that 9. 13(3) = 2:; m. For every positive integer n, let 10. En = UL Ai, and verify that

11. for every n, P(Bn) = Et, P(Ai), 12. for every n, IBn < IBn+1, 13. the sequence {IB,} is pointwise convergent, 14. limn +0 IBn = IB, 15. the sequence {XIBn} satisfies the hypotheses of the Monotone Convergence Theo- rem, 16. the sequence {XIBn} is pointwise convergent, and 17. limn - XIBn = XIB. Finally, show that 18. P(B) = Er, P(Ai) . NO