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

(a) $\displaystyle A_{r}\subseteq A_{s}$ if $\displaystyle r\leq s$,

(b) $\displaystyle \lambda (A_{s})=s$ for any $\displaystyle 0\leq s \leq \lambda(A)$.

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

$\displaystyle \lambda (A_{s})=\lambda ([0,s])=s$

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

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

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?