Pre-measure on a semi-ring

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Needs fleshing out urgently. Follows the doctrine of measure theory


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.