[分析] 兩題實變

[wyob (Go Dolphins)]

1.Let f be a bounded measurable function on E. Show that there are

  sequences of simple functions on E,{Hn} and {Jn}, such that {Hn}

  is increasing and {Jn} is decreasing and each of these sequences

  converges to f uniformly on E.



2.Let f be a measurable funtion on E that is finit a.e on E and

  m(E)<∞.For each e>0,show that there is a measurable set F contained

  in E such that f is bounded on F and m(E~F)<e.


第一題我想應該是要用Approximation Theorem

可是不知道怎麼找

第二題一點頭緒都沒有

感恩感恩

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→ doubleN     :1.用 Egorov's thm  2.用 Lusin's thm
→ wyob        :感謝樓上，不過你提到的兩個定理都在這個題目之後的
→ wyob        :章節，請問除了這兩個定理該怎麼寫比較好