This is just another answer trying to produce a set of finite measure to apply Egorov's Theorem to the sequence ${f_k}$.
You are using the book Measure and Integral by Richard L. Wheeden and Antoni Zygmund, me too, so I will refer some parts of that book.
Lemma 1. Suppose that $f\in L(\mathbb{R}^d)$, then $$\lim_{n\to\infty}\int \vert f\chi_{B(0,n)^c} \vert=0,$$
where $B(0,n)=\{\mathbf{x}\in \mathbb{R}^d:\Vert \mathbf{x}\Vert\lt n\}$ and the superscript $c$ is for complement.
Proof. Consider the sequence $\{f_k\}$ given by $$f_k=f\chi_{B(0,k)^c}.$$ Then $\vert f_k\vert\searrow 0$ and $\vert f_k\vert\leq \vert f\vert$ since $\vert f_k\vert\to 0$ as $k\to\infty$, and $\vert f\vert\in L$ by the Monotone Convergence Theorem (5.32(ii)) we get the desired result.
Lemma 2. Let $f\in L(E)$. Given $\epsilon\gt 0$, there exist $\delta \gt 0$ s.t. if $A\subseteq E$ and $m(A)\lt \delta$ then $$\int_A f\lt \epsilon.$$
The above Lemma is just the Theorem (7.1). The background for this theorem is at the beginning of the Chapter 7.
Solution to the exercise. Let $g\in L^q$. First note that if $\Vert g\Vert_q=0$ then $g=0$ a.e. and there is nothing to prove. Suppose that $\Vert g\Vert_q\gt 0$. Let $\epsilon\gt 0$. As a consequence of Fatou's Lemma, and Minkowski's Inequality, we get $\Vert f-f_k\Vert_p\leq 2M$.
By the Lemma 1, there exist $N\in \mathbb{N}$ s.t. $$\int_{B(0,N)^c} \vert g\vert^q\lt \left( \frac{\epsilon}{6M} \right)^q.$$
Let $B=B(0,N)$.
By Lemma 2, there exist $\delta\gt 0$ s.t. if $m(A)\lt\delta$ then $$\int_A \vert g\vert^q\lt \left( \frac{\epsilon}{6M} \right)^q.$$
Since $f_k\to f$ pointwise a.e. and $m(B)\lt \infty$, by Egorov's Theorem there exist $E\subseteq B$ such that $f_k\to f$ uniformly in $E$ and $m(B\setminus E)\lt \delta.$
Since $f_k\to f$ uniformly in $E$, there exist a $N'\in\mathbb{N}$ s.t. if $k\geq N'$ then $$\Vert f_k-f\Vert_{\infty,E}\lt \frac{\epsilon}{3\Vert g\Vert_q m(E)}.$$
Then if $k\geq N'$, we get
$$\begin{align*}
\int \vert f_k-f\vert\vert g\vert &= \int_{E} \vert f_k-f\vert\vert g\vert + \int_{B\setminus E} \vert f_k-f\vert\vert g\vert + \int_{B^c} \vert f_k-f\vert\vert g\vert\\
&= \int (\vert f_k-f\vert\chi_E)\vert g\vert + \int \vert f_k-f\vert\vert g\chi_{B\setminus E}\vert + \int \vert f_k-f\vert\vert g\chi_{B^c}\vert
\end{align*}$$
apply Hölder's inequality to the three terms
$$\begin{align*}
\int \vert f_k-f\vert\vert g\vert &= \Vert (f_k-f)\chi_E\Vert_p\Vert g\Vert_q + \Vert f_k-f\Vert_p\Vert g\chi_{B\setminus E}\Vert_q+\Vert f_k-f\Vert_p\Vert g\chi_{B^c}\Vert_q\\
&\leq \Vert f_k-f\Vert_{\infty,E}m(E)\Vert g\Vert_q + 2M\Vert g\chi_{B\setminus E}\Vert_q+2M\Vert g\chi_{B^c}\Vert_q\\
&\lt \epsilon.
\end{align*}$$
All this says that $\Vert f_kg-fg\Vert_1\to 0$.
Note that for $p=1$ the result fails. Take $f_k=k\chi_{[0,1/k]}$ and $g=\chi_{[0,1]}\in L^\infty$. Then $f_k\to 0$, $\Vert f_k\Vert_1=1$ for each $k$, and $\int fg=0$ but $\int f_kg=\int f_k=1$ for each $k$.
