Question

real Analysis

1). Let C be the intersection of all the Cn's. You must show C is closed, measurable, has positive measure, and contains no interval, i.e., if x is in C, every epsilon neighborhood of x contains points not in C.Analysis

i should have note 0<alpha<1

Start with the interval [0,1] and remove the middle open interval of length alpha/3 to form C1. Cn is then formed by removing the middle open interval of length alpha/3^n from each closed interval of C(n-1). Let C be the intersection of all the Cn's. You must show C is closed, measurable, has positive measure, and contains no interval, i.e., if x is in C, every epsilon neighborhood of x contains points not in C.

Answer 1

We first show that C is closed and measurable

Measure of C is followed from C being closed only

The second claim is not true as measure of this set is not positive if 0<α<1 rather it is zero as,

The last part is proved through contradiction as we have considered p has an open interval contained in C but latter shown it is not possible to prove the theorem

Hope it helps :-)