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  "body": "\n## Summary\n\nA premeasure is a countably additive size function defined only on a [[ring of sets|KWE1DBMN9B]].\nSuch a ring is closed under finite unions and differences, but not under countable unions.\nIt is the raw material for building a [[measure|PR11BZQQPG]].\nBy [[Carathéodory's extension theorem|Q8EXG7EKYW]] every premeasure extends to a measure on the generated $\\sigma$-algebra.\n\n## Explanations\n\n### L1 human\n\nOften one knows the size of simple sets, such as finite unions of intervals, before one knows the size of complicated sets.\nA premeasure records these sizes and asks that they add up correctly, even when a simple set is cut into countably many simple pieces.\nFor example, length on finite unions of half-open intervals $(a, b]$ is a premeasure.\nIf $(0, 1]$ is cut into the pieces $(1/2, 1]$, $(1/4, 1/2]$, $(1/8, 1/4]$, and so on, the lengths $1/2 + 1/4 + 1/8 + \\cdots$ add up to $1$.\n",
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