Provide step. by step solutions to the following attachment.this question is complete.

Problem 3. Let X be a set, (Y, Ey, my) be a measure space, and f : X - Y be a function. We define Ey C P(X) by Es = {E : f(E) e oy } and for each E E of we let HI(E) = HY(f(E)). a) Give an example where (X, Ey, my) is not a measure space. b) What property does the pre-image f-have that the image f does not have, which makes (X, Ey, uy) not always a measure space? c) Prove that if the function f : X - Y is one-to-one and f(X) e Ey then Ey is a o-algebra. d) Prove that if f : X - Y is one-to-one and f(X) E Ey then my is a measure on (X, E,). Problem 4. We now use Problem 3 to generate new measures on (R. BR). Recall that A is Lebesgue measure on R. Let f : R - R be a strictly increasing and right-continuous function. Recall that f is strictly increasing if a 0 and zo E R there exists o > 0 such that if 20