In Tao's book there is a sentence: "Here, there is an asymmetry (which ultimately arises from the fact that elementary measure is subadditive rather than superadditive): one does not gain any increase in power in the Jordan inner measure by replacing finite unions of boxes with countable ones" I cannot understand why for defining a measurability of a set we don't use the Jordan-like definition which makes a lot of sense. in Jordan measurability definition, we say a set is measurable if the outer Jordan measure equal the inner Jordan measure. Why in lebesgue definition we don't have this? also what does this sentence from Tao's book mean?



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