[Solved] Dynamical Homework 2

1.1 Section 2.3

Problem 1.

Suppose A is measurable. Then for any >), there is an open set G = G such that A G and (G A) < .

Problem 2.

Suppose A is a null set. Then for any > 0, there exists a sequence of intervals I_j such that A ^S_j=1 I_j and ^P_j=1 |I_j| < . Now consider A² = {a² : a A}. Note that if a I_k = (x,y) for some k, then

x < a < y

Problem 3.

Suppose A is any set, B is measurable, and (A4B) = 0. Note that A4B = (A B) (B A). Also note that (A B) (B A) = . So we have,

(A4B) = (() (B A))

= (A B) + (B A)

= 0

Thus, we have that (A B) = (B A). Since outer measure is non-negative, we must have that,

(A B) = 0 = (B A)

So we have that AB is a null set and is thus measurable. Now note that A = (AB) (A B) and that (A B) (A B) = . Since B is measurable, we have that there exists an open set G such that B G and (G B) <

Problem 6.

Problem 9.

1

1.2 Section 2.4 Problem 2.

Problem 4.

1.3 Section 2.5 Problem 1.

Problem 6.

Problem 7.

Problem 10.

Problem 13.