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Let $\left\{f_k\right\}_{k=1}^{+\infty}$ be an equi-integrable set of functions such that $f_k\to f$ in $L^{1}(\Omega)$, where $\Omega$ is a bounded domain in $\mathbb{R}^n$. Is it true that if we take any integrable function $g$ from $L^{1}(\Omega)$, for which $\left\{f_k\cdot g\right\}_{k=1}^{+\infty}\subset L^{1}(\Omega)$,$f\cdot g \in L^1(\Omega)$ then the family $\left\{f_k\cdot g\right\}_{k=1}^{+\infty}$ is equi-integrable?

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This is not true. Take $\Omega = (0,1/2)$ and \begin{align*} f_k(x) &= \sqrt{k} \, \chi_{(0,1/k)}(x), \\ f(x) &= 0, \\ g(x) &= \frac1{x \, \ln^2(x)}. \end{align*} Then, $f_k \to f$ in $L^1(\Omega)$ and, thus, $\{f_k\}$ is equi-integrable. Moreover, $g \in L^1(\Omega)$.

However, \begin{equation*} \| f_k \cdot g \|_{L^1(\Omega)} = \frac{\sqrt{k}}{\ln(k)} \end{equation*} is not bounded w.r.t.\ $k$. Thus, $\{f_k \cdot g\}$ is not equi-integrable.

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