sobolev space- notation

Question

if I have a sequence $u_k\in W_{1,p}$

what does it mean that :

" every subsequence of $\nabla u_k$ which converge in the sense of measure...."

what is a convergence is the sense of measure?

thank you

Answer 1

Means that every subsequence of $\nabla u_k$, denote as $\nabla u_{k_n}$ convergence to some $v$ in measure. i.e., $$ \lim_{n\to\infty}\mathcal L\{x\in \Omega, |\nabla u_{k_n}(x)-v(x)|\geq \epsilon\}=0 $$ for all $\epsilon>0$.

Although I am not sure whether $v\in L^p$ or not, depends on your sequence $u_k$.