[分析] Lebesgue measure

[tsaihohan (有人想一起下圍棋嗎)]

      Given a Lebesgue measurable subset A of R^n, we denote

      by │A│ the Lebesgue measure of A. Assume that {E_r:r屬於N}

      is a decreasing sequence of measurable subsets of R^n so

      that E_μ1包含E_ν whenever μ≦ν. We define ε=∩E_r, r€N.


      (a) Suppose that │E_r│ is finite for some r屬於N. Prove

      that │ε│=lim r→+∞│E_r│

      (b) Suppose that the equality │ε│=lim r→+∞│E_r│

      could be wrong when │E_r│=∞, for any r屬於N.

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推 plover      :(b) In R^1, define E_r = [r,∞).