Lebesgue Integrals proof

Verify 11-18.

Given 1. a measure space (9, .17 , P) and 2. a nonnegative, measurable function X on .Q. Show that 3. the function [A5 : .7: > [0, 00] E I> / X dP E is a measure on ([2, .7). That is, Show that 4. is countably additive, and 5. is not identically 00. First, verify that 6. is not identically 00. To establish the countable additivity of 13, let 7. 141,142, . . . be a sequence of pairwise disjoint members of .7: and 8. B 2 U2, A,. We wish to show that 9. RB) = 2.21%.). For every positive integer n, let 10. Bn 2 U?=1 A\" and verify that 11. for every n, P(Bn) = Et, P(Ai), 12. for every n, IBn XIBn = XIB. Finally, show that 18. P ( B) = Ex, P ( Ai )