Let $\Omega$ be an open bounded subset of $\mathbb{R}^n$ and let $ p\geq 1$. Let $(u_n)_n$ be a bounded sequence of $W_0^{1, p}(\Omega)$ and let $\Omega_n\subset\Omega$ be a subset of $\Omega$ which depends only 0n $n$ and such that $$meas(\Omega_{n})\longrightarrow 0 \quad \mbox{ as } n\to +\infty.$$ Could I conclude that $$\int_{\Omega_{n}} \vert\nabla u_n\vert^{p} dx\longrightarrow 0 \quad \mbox{ as } n\to +\infty?$$ If not, what additional assumptions I need?
Could anyone please help? Thank you in advance!