I am working through Evan's book on Weak Convergence Methods for Nonlinear PDE. He assumes that $U$ is an open bounded smooth subset of $\mathbb{R}^n$ and that $1<q<n$. In ($\S$D) concerning Measures of Concentration he assumes that $f_k$ is a sequence of functions in $L^q(U)$ that converge weakly to $f$ but not strongly to $f$. He has the following statement under this section :
Secondly, observe that even if we somehow know additionally $f_k \rightarrow f\ a.e.$ in U; so that wild oscillations are excluded, we still cannot legitimately deduce strong convergence in $L^q(U)$. The obstruction is that the mass of $\vert f_k-f\vert^q$ may somehow coalesce onto a set of zero Lebesgue measure. This is the problem of concentration.
I am trying to understand what he means by
$\dots$ the mass of $\vert f_k-f\vert^q$ may somehow coalesce onto a set of zero Lebesgue measure $\dots$
Does he mean that as $k\rightarrow\infty$ that (somehow): \begin{equation} \mathcal{L}^n(\{x\in U\vert\ \vert f_k(x)-f(x)\vert ^q\neq 0\})\rightarrow 0, \end{equation} where $\mathcal{L}^n$ is the n-dimensional Lebesgue measure?
I would guess that he means something like \begin{equation} \mathcal{L}^n(\{x\in U\vert\ \vert f_k(x)-f(x)\vert ^q > \epsilon\})\rightarrow 0 \end{equation} for all $\epsilon > 0$.
An example of such behaviour is the sequence of functions in $L^2(0,1)$ $$ f_n(x) = \begin{cases} \sqrt n & 0 < x < 1/n, \\ 0 & \text{else}. \end{cases} $$ Then, $f_n \rightharpoonup 0$ and $f_n(x) \to 0$ for all $x$, but $\|f_n\|_{L^2} = 1$.