My question refers to Federer's book on geometric measure theory, 2.10 Caratheodory's construction around p.169. Basically the question is: "Why is it possible to apply Caratheodory's criterion on the measure $\psi=\lim_{\delta\rightarrow 0}\phi_\delta$, when it does not apply for any $\delta >0$?"
Consider the open set $(0,\frac{\delta}{2})\cup(\frac{\delta}{2},\delta)$ and the gauge $\operatorname{diam}(A)^\frac{1}{2}$. The set is not $\phi_\delta$ measurable, since Caratheodory's criterion does not hold: $$\phi_\delta\Big((0,\frac{\delta}{2})\cup(\frac{\delta}{2},\delta)\Big)=\delta^\frac{1}{2}<2 \Big(\frac{\delta}{2}\Big)^\frac{1}{2}=\phi_\delta((0,\frac{\delta}{2}))+\phi_\delta((\frac{\delta}{2},\delta))$$
I am struggling with the right intuition. I see, that "problems" occur for two disjoint open sets $A,B$ with $\operatorname{dist(A,B)=0}$, but in this case I am still wondering how to calculate $\psi(A\cup B)$ in the right way, i.e. as the limit of $\phi_\delta(A\cup B)$.