Definition. Let $\lambda^*$ denote the Lebesgue outer measure on $\mathbb {R}$. A set $E \in P(\mathbb{R})$ satisfies the Carathéodory condition if: $$ \forall T \in P(\mathbb{R}), \space\space\lambda^*(T) = \lambda^*(T \cap E) + \lambda^*(T \cap E^c)$$
I can not find a set $E$ such that: $$ \textrm{ for some } T \in P(\mathbb{R}), \space\space\lambda^*(T) < \lambda^*(T \cap E) + \lambda^*(T \cap E^c)$$
May you help me?
Thanks!
Let $E$ be a Bernstein set in $T = [0,1]$. Then $\lambda^*(T) = 1$, $\lambda^*(T\cap E) = 1$, $\lambda^*(T\cap E^c) = 1$.
Any non-measurable set $E$ will do. In fact, Lebesgue's definition of "measurable" for unbounded sets $E$ was $$ \lambda^*(T) = \lambda^*(T \cap E) + \lambda^*(T \cap E^c) $$ for all intervals $T = [a,b]$. Carathéodory, when he wanted to generalize to other situations, noted that equivalently you get the same thing if you say "all sets $T$".
