If A is a Lebesgue measurable subset of [0,1] such that \lambda (A)>0. I need to show that there are Lebesgue measurable subsets A_{s},0\leq s \leq \lambda(A) so that

(a) A_{r}\subseteq A_{s} if r\leq s,
(b) \lambda (A_{s})=s for any 0\leq s \leq \lambda(A).

I started with A_{s}=A\cap [0,s] and I can verify (a) easily. But I only have (b) if [0,s]\subseteq A since
\lambda (A_{s})=\lambda ([0,s])=s

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

How can I modify my construction of the set A_{s}?

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?