I'm solving exercise 8.11 in Wheeden & Zygmund's measure and integral: 
Does the symbol "$f_k \rightarrow f$ in $L_p$" by definition means $f_k$ in $L_p$? Or just $\lim_{k\rightarrow\infty} (\int |f_k-f|^p)^{1/p} = 0$ with $f_k$ and $f$ mere measurable?
If $f_k$'s are in $L_p$ then it's easy, but I don't know how to do it if they aren't.
It is generally implied that $f_k$ and $f$ are in $L^p$ when writing $f_k\rightarrow f$ in $L^p$. In fact the statement is false if $f_k$ is not in $L^p$. Consider the sequence $f_k:(0,1)\rightarrow \mathbb{R}$ defined by $f_k(x)= 1/x$ and the sequence $g_k(x) = 1/k$ on the same domain. Then let $f(x)=1/x$ and $g(x)=0$.
Then clearly $\int_{(0,1)} | f_k-f|^p = 0$; however \begin{equation} \int_{(0,1)} |f_kg_k-fg|^p = \int_{(0,1)} \left|\frac{1}{kx}\right|^p =\infty \end{equation}