Why is the lebesgue measure equal to K?

Question

I have the following question. Let $\lambda$ be the lebesgue measure. Let $O$ be open and dense in $[0,K]$. Why is $$\lambda(O)=K?$$

Answer 1

As zhw. has pointed out in a comment, this isn't true, and here is a counterexample: Let $q_1,q_2,...$ be enumeration of all rational numbers in the interval $[0,1]$. Take $O$ to be the open set which is union of open intervals $(q_i-4^{-i},q_i+4^{-i})$. This set clearly contains all the rationals of $[0,1]$, so must be dense in $[0,1]$, while the measure of $O$ is at most the sum of lengths of the intervals, which is $\sum_{i=1}^\infty\frac{2}{4^i}=\frac{2}{3}<1$.