Assume that $\mu$ is a Borel measure on $[0,1]$ and let $\lvert\, \cdot\, \rvert$ be the Lebesgue measure on $[0,1]$. Suppose that for any Borel set $A\subset[0,1]$ with $\lvert A\rvert=\frac{1}{2}$ we have $\mu(A)=\frac{1}{2}$. Prove that $\mu=\lvert\,\cdot\,\rvert$.
This is my homework and I do not even know how to start. I suppose I should see the connection with the $\pi-\lambda$ method, but I cannot find any $\pi$–system in this problem. Does the collection of all Borel sets of the Lesbesgue measure $\frac{1}{2}$ generate the Borel $\sigma$–algebra? I would be grateful for your help.