I am studying the calculus of variations. Many interesting results arise from the theory of Young measures. For a greenhorn (as I am) an intuition behind Young measures is hidden, because many texts state (without introduction) just the main (Ball's) theorem about existence of Young measures and then continue on applications.
My question - Is there any intuitive insight on Young measures? What are they measuring (concentration?, oscillations?, ???)? Is there any connection with probability?
Young measure is introduced in order to describe the weak$^*$ $L^\infty(\Omega)$-convergence, i.e., of a sequence $\{u_n\}\subset L^\infty(\Omega)$, with $$ \int_\Omega u_n\,v\,dx\to\int_\Omega u\, v\,dx, \quad \text{for every $v\in L^1(\Omega)$}, $$ by a probability-measure-valued function $\mu_x$, $x\in\Omega$. This probability-measure-valued function's main use is that it can describe where $f(u_n)$ converges weakly, where $f$ bounded: $$ f(u_n)\to \bar f, $$ weak$^*$, where $\bar f(x)=\int_{\mathbb R} f(\lambda)\,d\mu_x(\lambda)$.