Brief Overview:

Studying for final, doing random problems out of book for practice.

Please do not just give me the answer, that won't do me any good, rather look at my ideas and tell me what direction to head in or how to better understand the concepts the problem is envoking, thanks.

Let S be the subset of x-axis consisting of the union of the open interval of length 1/4 centered at 1/2, and the open intervals of length 1/16 centered at 1/4 and 3/4, the open intervals of length 1/64 centered at 1/8, 3/8, 5/8, and 7/8, and so forth. Let U = S x (0,1) be the union of the open rectangles of height 1, based on these intervals. Thus U is the union of one rectangle of area 1/4, two rectangles of area 1/16, four rectangles of area 1/64, ... some of which overlap.

a. Show that U is an open subset of of the unit square on

R = [0,1] x [1,0]

my solution:

Open intervals with the largest centers are of the form:

$\displaystyle (\tfrac{2^{k} - 1}{2^{k}} - \tfrac{1}{2(4^{k})} , \tfrac{2^{k} - 1}{ 2^{k} } + \tfrac{1}{2(4)^{k}} )$

$\displaystyle \tfrac{2^{k} - 1}{2^{k}} + \tfrac{1}{2^{2k+1}} \leq 1 - \tfrac{1}{2^{k}} + \tfrac{1}{2^{2k+1}} < 1 - \tfrac{1}{2^{k}} + \tfrac{1}{2^{k}} = 1 $

The argument for open intervals with the smallest centers is similar except for the form which is

$\displaystyle (\tfrac{1}{2^{k}} - \tfrac{1}{2(4^{k})}, \tfrac{1}{2^{k}} + \tfrac{1}{2(4^{k}) } )$

b. Show that the inner area of U is less than 1/2.

I dont see how to get the inner area of U without double counting the area...as the problem said, some rectangles intersect. And I dont see any trick to indirectly get the inner area.

c and d, talk about the outer areas.

I am confused on how to think about outter and inner areas in this problem. All I really know are the definitions of inner and outter area, but still I dont see how to get past that probem of double counting or how I might apply the properties which occur with the characteristic function when U is measurable.

Help is much appreciated.