The course addresses the Lebesgue measure of subsets of real numbers. The text book is "Real Analysis" of Royden and Fitzpatrick.
Here you can find a proof of the existence of a non-measurable set.
Here you can find two exercises. Please, also look at exercises 1 and 2 at page 31 of the book.
These are the solutions. And these are the solutions of exercises 1 and 2 of page 31 of the book.
Here you can find some exercises on the topology of the set of real numbers.
And these are the solutions.