The question is quite "simple". Let $\lambda^*$ denote the usual Lebesgue outer measure on $\mathbb R.$ Let $E\subseteq [0,1]$ be a non-measurable subset. Do we always have $$ \lambda^*(E) +\lambda^* ([0,1]\backslash E) >1? $$ Are there examples of non-measurable sets such that equality $\lambda^*(E) +\lambda^* ([0,1]\backslash E) =1$ holds?

Although this is very easy to state, it is very hard to think of what examples do we have. After all, the construction of a non-measurable set along is not that easy. The Vitali set does not provide the example asked for above.