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    Real analysis : Measurable Functions

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    Q1. If 0<=a_1<=a_2<=a_3<=....,( 1,2,3 are the subscripts of a)
    0<=b_1<=b_2<=b_3<=......(1,2,3 are the subscripts of b)
    and a_n --> a and b_n -->b
    Then prove that a_n*b_n -->a*b

    Q2.Let f: R --> R be monotonically increasing, i.e.
    f(x_1)<= f(x_2) for x_1< = x_2.
    Show that f is measurable.
    Hint: You may extend f to f':[-infinity,infinity]-->[-infinity,infinity]and
    show that (x_alpha,infinity] is contained in f'^-1((alpha,infinity])
    is contained in [x_alpha,infinity],where x_alpha=inf{x:f'(x)>alpha}

    Please respond with Complete proof/solution.
    NOTE: f' means a bar on the head of f

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    Question 1: Proof:
    Since and , then for enough big , we have . This implies . In another word, when is big enough, is bounded by . ...

    Solution Summary

    Measurable functions are investigated. The solution is detailed and well-presented.