If is a Lebesgue measurable subset of [0,1] such that . I need to show that there are Lebesgue measurable subsets so that

(a) if ,

(b) for any .

I started with and I can verify (a) easily. But I only have (b) if since

But (b) is not true if [0,s] is not a subset of .

How can I modify my construction of the set ?

The s in this problem is a bit confusing for me as it is a measure and subscript at the same time. I don't really understand the requirement in (b). Any idea?