I'm taking a course in which Young measures are introduced for oscillation and concentration. I have understood the examples the lecturer has given us for concentration Young measures, but cannot get my head around the ones for oscillation.
Here is an example:
Let $v_j:=\sin(jx)$, let $\Omega=(0,1)$ and let $\mathbb{E}_1$ be the space of "1-admissible integrands", that is the space of $\Phi \colon \Omega \times \mathbb{R}\to\mathbb{R}$ such that $\lim_{t\to\infty}\frac{\Phi(x,tz)}{t}$ exists for $z\in\{+1,-1\}$.
The functions $v_j$ may be viewed as 1-Young measures, which are viewed as elements of the dual space to $\mathbb{E_1}$, under the duality relation \begin{align*} \langle\langle v_j,\Phi\rangle\rangle&:=\int_\Omega\int_\mathbb{R}\Phi(x,z)d\delta_{v_j(x)}dx \\ &=\int_{\Omega}\Phi(x,v_j(x))dx \end{align*} We claim that the $v_j\to v$ for some $v$ as Young measures in the sense that they tend weak* in $\mathbb{E}_1^*$. That is, we claim that there exists a family $v=(v_x)_{\{x\in\Omega\}} $ where each $v_x$ is a probability measure on $\mathbb{R}$ such that for any $\Phi\in\mathbb{E}_1$ \begin{align*} \langle\langle v_j,\Phi\rangle\rangle &\to \langle\langle v,\Phi\rangle\rangle \\ &:= \int_\Omega\int_\mathbb{R} \Phi(x,z)dv_x(z)dx \end{align*}
as $j\to\infty$. Our lecturer claims that this Young measure $v$ is given by the pushforward of the Lebesgue measure restricted to $(0,2\pi)$, under the map $\sin(\cdot)$, divided by $2\pi$, at all points $x\in\Omega$. That is $$v_x=\sin(\cdot)_*\left(\left(\frac{1}{2\pi}\right)\mathcal{L}\llcorner(0,2\pi) \right) $$ Where the pushforward measure is defined as $f_*(\mathcal{L})(A)=\mathcal{L}(f^{-1}(A))$. They also claim that this follows from the Riemann-Lebesgue lemma.
Is anyone able to explain why this is the limit? It is not obvious to me how the integrals converge, let alone to what is stated.
Many thanks for any advice, A.