3. Consider the Cantor set D formed by deleting the middle subinterval of length 4-* from each re...
3. Consider the Cantor set D formed by deleting the middle subinterval of length 4-* from each remaining interval at step k. (a) Prove that the length of the D is 1/2. Thus D is a fat fractal. (b) What is the box-counting dimension of D? (c) Let be the function of [0,1] which is equal to 1 on D and 0 elsewhere. It is the limit of functions which are Riemann integrable. Note that f is not Riemann integrable. What is the value of any lower Riemann sum for f (The lower sum is the sum of box areas using the infimum of the function J() in each subinterval(A-))
3. Consider the Cantor set D formed by deleting the middle subinterval of length 4-* from each remaining interval at step k. (a) Prove that the length of the D is 1/2. Thus D is a fat fractal. (b) What is the box-counting dimension of D? (c) Let be the function of [0,1] which is equal to 1 on D and 0 elsewhere. It is the limit of functions which are Riemann integrable. Note that f is not Riemann integrable. What is the value of any lower Riemann sum for f (The lower sum is the sum of box areas using the infimum of the function J() in each subinterval(A-))