Prove that there exists no Lebesgue-measurable set $A$ such that for every interval $I$ the Lebesgue measure of $A\cap I$ is half the length of $I$.

Question

So I know the following problem was solved on this website. However, I found a solution to it that seems too simple to be right. Therefore, I was hoping you could check my proof and point out the error, or its lack if I am lucky :P.

Let $I=[a,b]$ then $I^C=(-\infty,a)\cup(b,\infty)$. By carethodory criterion we know that $\lambda(I)=\lambda(A\cap I^C)+\lambda(A\cap I)$ then $\lambda(I)/2=\lambda(A\cap I^C)$ However the left handside is finite and the right handside is infinite. Therefore, this is impossible and such an $A$ cannot exist.

Answer 1

You claim that $\lambda(I) = \lambda(A \cap I^c) + \lambda(A \cap I)$. There's your (first) error.