How can I prove that $f_n$ converges pointwise to $f$ such that

$$ f_n=\begin{cases} f,&\text{ in }A_n\\ 0,&\text{ in }A_n^c \end{cases}$$ Where $A_n=\{ x\in X\,:\, 1/n \leq f \leq n, n\in\Bbb \}$ and $f$ is positive and in $L^p$?


closed as off-topic by Kavi Rama Murthy, RRL, Shailesh, Ak19, Adrian Keister Jun 19 at 15:15

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$$\|f_n-f\|^p_p=\int_{A_n}|f_n-f|^p d \mu+\int_{A_n^c}|f_n-f|^p d \mu=0+\int_{A_n^c}|f|^p d \mu.$$ Now split the last integral into two cases $f>n$ and $0<f<\frac{1}{n}$. Think, one of the integrals is over a set of measure zero (which one and why?)


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