We are currently covering the last chapter of baby Rudin where he quickly covers the basics of Measure Theory. I am having a hard time understanding under what circumstances does the outer measure equal the Lebesgue measure. From looking at other sources, it appears that this occurs when the set in question is measurable, but I fail to understand how that can be determined when all we have is the outer measure. If someone could help clarify this to me that would be greatly appreciated.
2026-03-27 23:02:03.1774652523
Measure Theory question about Outer measure vs measure
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The Lebesgue measure is the Lebesgue outer measure restricted to the "Lebesgue measurable" sets. The Lebesgue measurable sets are a large class of sets on which the Lebesgue outer measure is a measure. See https://en.wikipedia.org/wiki/Carath%C3%A9odory%27s_extension_theorem