If $g_k(x) = k \log\big(1 + \frac{g(x)^2}{k^2}\big) \text{ with} \ g \in L^1([0,1])$, I know the $$\lim\limits_{k}\int_0^1 g_k = 0$$ but my issue is finding the control function.
Using the properties of the log, we can show that $g_k$ is dominated by $\frac{g^2}{k}$. Problem: it is not guaranteed that $g^2 \in L^1([0,1])$?
edit: I think it is bounded by $|g|$.
Since for $x$ positive, $\log(1+ x^2/2 ) \le x$, which follows from $1+x^2/2 \le e^x$ (look at the Taylor series for $e^x$),
$$ g_k(x) \le k \sqrt{2}\frac{|g(x)|}{k} =\sqrt{2} |g(x)|. $$ Hence when $g$ is integrable, the sequence $g_k$ is bounded by an integrable function.