If $(X,d)$ is a metric space, say a function $\tau$ on some class $\mathscr{C}$ of subsets of $X$ is a pre-measure, if $\emptyset \in \mathscr{C}$, $\tau(\emptyset)=0$ and $0\le \tau(C)\le +\infty$ for all $C\in \mathscr{C}.$

If $\tau$ is a pre-measure on some class $\mathscr{C}$ of subsets of $X$, then the set function
$\mu(E)=\inf\left\{\sum_{i=1}^\infty\tau(U_i):U_i\in \mathscr{C}, E\subset \cup_{i=1}^\infty U_i\right\}$
is a measure on $X.$ The measure is called **Method I measure**. For more detail, see C.A. Rogers's book titled *Hausdorff measure*.

One claims that "Borel sets are not in general measurable with respect to measures from Method I constructions" in a literature. Can someone give an example? Thanks.