Let $ M\subseteq 2^X$ be a collection of subsets of $X$ and $\rho:M\to [0,\infty]$ be a set function such that $\emptyset,X\in M$ and $\rho(\emptyset)=0$. The set function $\rho$ induces an outer measure given as $$\mu^*(A)=\inf\left\{\sum_{i=1}^\infty \rho(E_i)\ \biggr| \ E_i\in M\text{ and }A\subseteq \bigcup_{i=1}^\infty E_i\right\}.$$ This is the "standard" construction of an outer measure on some set $X$. Carathéodary's criterion can be used to define a measure $\mu$ using this outer measure $\mu^*$. Is there an example where this induced measure $\mu$ disagrees with the original set function $\rho$? $$ \exists E\in M\text{ s.t }\rho(E)\stackrel{?}{\neq}\mu(E)$$
Royden and Fitzpatrick motivate (what they refer to as) the Carathéodory-Hanh Theorem and pre-measures in Section 17.5 by asking what properties must $\rho$ and $M$ satisfy such that $\rho$ and $\mu$ agree (where they're both defined), but they don't provide an example where this equality doesn't hold in the absence of the additional assumptions that comprise the Carathéodory-Hanh Theorem.