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Math Help - convergence in Lp space

  1. #1
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    convergence in Lp space

    If (f_n) is a sequence of characteristic function of sets in X and if (f_n) converges to f in L_p(X,M_x,m),show that f is (almost everywhere equal to) the characteristic function of a set in X.

    Since (f_n) converges in L_p,then it is convergent in measure.I try to connect things in this way but I am stuck.Can anyone give me any hints to proceed?
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  2. #2
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    Quote Originally Posted by problem View Post
    If (f_n) is a sequence of characteristic function of sets in X and if (f_n) converges to f in L_p(X,M_x,m), show that f is (almost everywhere equal to) the characteristic function of a set in X.
    Since each f_n is real, it follows that f is real. If E is any subset of X with finite measure, and g is the characteristic function of E, then g\bigl|f_n-\tfrac12\bigr| is in L_p(X,M_x,m), and g\bigl|f_n-\tfrac12\bigr|\to g\bigl|f-\tfrac12\bigr| in L_p(X,M_x,m). But \bigl|f_n-\tfrac12\bigr| is identically equal to 1/2. It follows that \bigl|f-\tfrac12\bigr| = \tfrac12 almost everywhere on E, and so f takes the value 0 or 1 almost everywhere on E (and hence almost everywhere on X). Thus f is almost everywhere equal to the characteristic function of a set in X.
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