[分析] 實變數題

[FANggot (憨吉)]

我想要問的是 non-measurable sets.

在Royden 4th Ed.中 有一題題目如下


Let E be a nonmeasurable set of finite outer measure.

Show that there is a G-delta set G that contains E for which

m*(E) = m*(G) while m*(G\E) > 0


在這裡m*是outer measure 的notation.


我知道怎麼建構一個G-delta set G 使得 m*(E) = m*(G), 但是

我不太瞭解要怎麼使 m*(G\E) > 0


我想說用反證法:

要是每一個滿足 m*(E) = m*(G) 的 G-delta set G (包含 E),

也都滿足 m*(G\E) <= 0,

那麼m*(G\E)=0.

Then by outer approximation by G-delta sets, E is measurable. -><-





我好奇的是上面的論證對嗎?

我好困惑啊~~




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推 math1209    :yes. 就是這樣子.