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Thread: convergence in Lp space

  1. #1
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    convergence in Lp space

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

    Since $\displaystyle (f_n)$ converges in $\displaystyle L_p$,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 $\displaystyle (f_n)$ is a sequence of characteristic function of sets in $\displaystyle X$ and if $\displaystyle (f_n)$ converges to $\displaystyle f$ in $\displaystyle L_p(X,M_x,m)$, show that $\displaystyle f$ is (almost everywhere equal to) the characteristic function of a set in $\displaystyle X$.
    Since each $\displaystyle f_n$ is real, it follows that $\displaystyle f$ is real. If $\displaystyle E$ is any subset of $\displaystyle X$ with finite measure, and $\displaystyle g$ is the characteristic function of $\displaystyle E$, then $\displaystyle g\bigl|f_n-\tfrac12\bigr|$ is in $\displaystyle L_p(X,M_x,m)$, and $\displaystyle g\bigl|f_n-\tfrac12\bigr|\to g\bigl|f-\tfrac12\bigr|$ in $\displaystyle L_p(X,M_x,m)$. But $\displaystyle \bigl|f_n-\tfrac12\bigr|$ is identically equal to 1/2. It follows that $\displaystyle \bigl|f-\tfrac12\bigr| = \tfrac12$ almost everywhere on $\displaystyle E$, and so $\displaystyle f$ takes the value 0 or 1 almost everywhere on $\displaystyle E$ (and hence almost everywhere on $\displaystyle X$). Thus $\displaystyle f$ is almost everywhere equal to the characteristic function of a set in $\displaystyle X$.
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