23/02/16 19:55:21.47 aVW5JpNj.net
Sheldon Axler著『Measure, Integration & Real Analysis』
|A| を A の外測度とします。
b - a ≦ |[a, b]| が自明かどうかについて以下のように書いています：
|[a, b]| ≦ b - a
Is the inequality in the other direction obviously true to you? If so, think again,
because a proof of the inequality in the other direction requires that the completeness
of R is used in some form. For example, suppose R was a countable set (which is not true,
as we will soon see, but the uncountability of R is not obvious). Then we would have
|[a, b]| = 0 (by 2.4). Thus something deeper than you might suspect is going on with
the ingredients needed to prove that |[a, b]| ≧ b - a.
解説が素晴らしすぎます。日本語のルベーグ積分の本は何なんだと思ってしまいますよね。