Pre-measure on a semi-ring
As per the doctrine, a pre-measure on this site refers to measure on a ring of sets. A pre-measure on a semi-ring of sets is a precursor to a pre-measure. We can uniquely extend a pre-measure on a semi-ring to a pre-measure.
We then extend the pre-measure to an outer measure and go from there. This simplifies obtaining a measure as we can "go through" a pre-measure to get there, so we need only show a pre-measure on a semi-ring can be extended to a pre-measure.