This is the second part of the Exercise 2 in Chapter 8 of Measure and Integral by Zygmund and Wheeden:
Show also that for real valued $f\notin L^p(E)$, there exist a function $g\in L^q(E)$, $\frac{1}{p}+\frac{1}{q}=1$, s.t. $fg\notin L^1(E)$. (Construct $g$ of the form $\sum a_kg_k$ for appropriate $a_k$ and $g_k$ satisfying $\int_E fg_k\to \infty$.)
In the following the integrals are taken over $E$.
Trying to follow the hint, if we assume $1\lt p\lt\infty$ note that $a_k=k^{-(q+1)}$, $b_k=k$ satisfies $\begin{cases} \sum a_kb_k\lt\infty\\ \sum a_kb_k^q=\infty\end{cases}$ In view of $\begin{align*} &\Vert g\Vert_q=\Vert \sum a_kg_k\Vert_q\leq \sum a_k\Vert g_k\Vert_q\\ &\Vert fg \Vert_1=\int fg =\sum a_k\int fg_k \end{align*}$ we just need $g_k$ such that $\Vert g_k\Vert_q=k$ and $\int fg_k=k^q$. Suppose that there exist a set $A_k\subseteq E$ such that $\int_{A_k} \vert f\vert^p=k^q$, then the sequence given by $g_k=\vert f\vert^{p-1}\chi_{A_k}=\vert f\vert^{p/q}\chi_{A_k}$ will do the job. I need to justify the existence of the $A_k$s. So, I'm looking for something like the following.
Let $E\subseteq \mathbb{R}^d$. Let $f$ a function with $\int_E \vert f\vert=\infty.$ My question is:
Given $t\in[0,\infty[$, does there exist a set $A_t\subseteq E$ s.t. $\int_{A_t}\vert f\vert=t?$
Let $C_t$ the cube around the origin with volume $t\geq 0$. I was trying to prove that the function $F:[0,\infty[\to\mathbb{R}\cup\{\infty\}$ given by $F(t)=\int_{C_t\cap E}\vert f\vert$ is continuous. My idea seems to have no future, because for example $f$ could be $\infty$ in any subset of positive measure.
If this is not possible, how can I approach this problem?