Let $f_n \in L^p(B)$ be a sequence where $B$ is some ball in $\mathbb{R}^n$. Assume that $\|f_n\|_{L^p(B)} \rightarrow 0$ when $n\rightarrow \infty$, then for any $\phi \in C^\infty_0(B)$, applying Hölder we get $$ \int_B f_n \phi \le \|f_n\|_{L^p(B)}\|\phi\|_{L^{p/(p-1)}(B)}. $$ Thus, $\int_B f_n \phi \rightarrow 0$ when $n\rightarrow \infty$.
Does the result remains true if we replace the condition $\|f_n\|_{L^p(B)} \rightarrow 0$ by $\|f_n\|_{L^p_w(B)} \rightarrow 0$? Here, $\|f\|_{L^p_w(B)}$ is the infimum of all constant $C$ such that $$ \mathcal{L}^n \{ x \in B: |f(x) | > \alpha \} \le \dfrac{C^p}{\alpha^p} \quad (\forall \alpha >0)$$ is the weak $L^p$-norm of $f$ in $B$. This is. Do we have $\int_{B} f_n \phi \rightarrow 0$ when $n\rightarrow \infty$?
Assume that $p>1$. Let $1<q<p$ and $f\in L^{p,\infty}$. Then $$\int_B|f(x)|^q\mathrm dx=q\int_0^{+\infty}s^{q-1}\mathcal L\{t:|f(t)>s\}\mathrm dx\leqslant q\lVert f\rVert_{p,\infty}\left(\int_{1}^{\infty}s^{q-p-1}\mathrm ds+\mathcal L(B)\right).$$ In particular, the inclusion $L^{p,\infty}\to L^q$ is continuous.
In this context, we have $\lVert f_n\rVert_{L^q}\to 0$, hence we can use the same argument as in the OP.