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Math Help - measurability

  1. #1
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    measurability

    I have a question. say, let <f_n> be a sequence of measurable functions with the same domain. and let h(x)=sup\{f_1 (x),...,f_n (x) \}.
    Then \{x:h(x)> \alpha \}= \bigcup_{i=1}^n \{x:f_i (x)> \alpha\}?
    i do not see why they are equal. to me the right side is bigger. please help me understand.
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  2. #2
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    Well, since h(x) is the supremum over a finite set it is equal to some element from that set so if \alpha < h(x)=f_j(x) for some j. Now using the fact that h(x)\geq f_i(x) for all i we get the other direction so the sets are equal.
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