In my road to understand microlocal defect measures, at the beginning of Gerard's article Microlocal defect measures, there is an statement about (an example of) defect measures where I am struggling.
The context is the following. Let $\Omega$ be an open set in $\mathbb{R}^d$ and let $(u_n)_n$ be a bounded sequence in $L_{loc}^2(\Omega)$ which converges to $u\in L_{loc}^2(\Omega)$ in the sense of the distributions. By identifying $L^1$ with a subspace of the space of Radon measures, a weak-$*$ argument allows us to deduce that $$|u_n-u|^2\rightharpoonup \nu$$ where $\nu$ is a Radon measure, called the defect measure associated to $(u_n)_n$.
To exemplify that the defect measures does not capture some important properties, the sequence $u_n(x)=\exp(inx\cdot \xi)$, for some $\xi\in \mathbb{R}^d\setminus\{0\}$, is examined. This sequence is bounded in $L_{loc}^2(\mathbb{R}^d)$, $u_n\rightharpoonup 0$ in the sense of the distributions and does not converge strongly in $L_{loc}^2$. The defect measure $\nu$ associated to this sequence is the Lebesgue measure, for any $\xi\in \mathbb{R}^d$.
My question is: how do I find that, indeed $\nu=\lambda$? I've been thinking about it, but I did not be able to unravel the statement. Any hint is well received!