Lebesgue Integrals

From Exercises 14: Nos. 11-18.

Exercises 14 Math 250 October 13, 2020 Given 1. a measure space (9, f, P) and 2. a nonnegative7 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