# Thread: convergence in Lp space

1. ## 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?

2. Originally Posted by problem
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 $X$.