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The Princeton Companion to Mathematics by Imre Leader, June Barrow-Green, Timothy Gowers

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III.55 Measures

To understand measure theory, and to see why it is useful and important, it is instructive to start with a problem about lengths. Suppose that we have a sequence of intervals in [0, 1] (the closed interval from 0 to 1), of total length less than 1. Can they cover [0, 1]? In other words, given intervals [a1, b1] [a2, b2], . . . , with Σ(bn - an) < 1, is it possible that their union equals [0, 1]?

One is tempted to answer “no, as the total length is too small.” But this is just to restate the question. After all, why should “total length less than 1” actually imply that the intervals cannot cover [0, 1]? Another tempting answer is to say “just rearrange the intervals so that they go from the left to the right, and then we never ...

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