Assume that q is a non-negative function defined on the unit interval(0.1) and that the integral of q is small. That is and where is a small positive number. We want to study the size of the set where...

Assume that q is a non-negative function defined on the unit interval(0.1) and that the integral of q is small. That is
and

where is a small positive number. We want to study the size of the set where q is “fairly” large. To this end, consider the set

We will assume that q is such that S consists of a countable disjoint union of intervals, i.e.,

(b) Solve the problem (8.1)–(8.3) in the case of

(c) Solve the problem (8.12)–(8.14) in the case of

(d) Compute

(e) Compute

(f) Define the set S by