About the convergence of a sequence in $L^1$

Question

Suppose that $f_n$ is a sequence of nonnegative functions such that $\int f_n d\mu=1$ for all $n$, and $f_n\to f$ in $L^1$. Let $p>1$. Is it then true that $f_n^p\to f^p$ in $L^1$?

Answer 1

This is false. Because as Chris remarks above, $f_n^p$ needn't be in $L^1$.

For example: If the concerned measure space is $\mathbb R$ with the Borel $\sigma$-field and the Lebesgue measure and $f$ is as defined below:$$\begin{cases}x<0&f(x)=0\\0<x\le1&f(x)=1\\1+\frac1{2^{p+1}}+...\frac1{(n-1)^{p+1}}<x\le1+\frac1{2^{p+1}}+...\frac1{n^{p+1}}&f(x)=n\\1+\frac1{2^{p+1}}+...\frac1{n^{p+1}}+...<x&f(x)=0\end{cases}$$

So on an interval of length$\displaystyle \frac1{n^{p+1}}$, $f$ has value $n$, so the $L^1$ norm is the sum $\sum\frac1{n^p}$ which converges for $p>1$.

But $f^p$ is $n^p$ on the respective intervals. So its $L^1$ norm is $\sum\frac1n$ which diverges!