Are there measures that can be obtain in a similar way to Lebesgue measure?

Question

For constructing Lebesgue measure, first we definite the Lebesgue outer measure and we use Caratheodory's theorem for restricting this outer measure to a measure. Are there more non-trivial measure that can be constructed in such a way? I mean, is there a non-trivial measure $\mu$ over a measurable space $(X,\mathcal{A})$, different from Lebesgue measure, which is constructed by first defining an outer measure and after aplying Caratheodory's theorem for restricting this outer measure to a measure.