The Young Measure Representation for Weak Cluster Points of Sequences in $M$-spaces of Measurable Functions
Volume 56 / 2008
Abstract
Let $\langle X, Y\rangle$ be a duality pair of $M$-spaces $X,Y$ of measurable functions from ${\mit\Omega}\subset\mathbb R^n$ into $\mathbb R^d$. The paper deals with $Y$-weak cluster points $\overline{\phi}$ of the sequence $\phi(\cdot,z_{j}(\cdot))$ in $X$, where $z_j\colon{\mit\Omega}\rightarrow\mathbb R^m$ is measurable for $j\in \mathbb{N}$ and $\phi\colon{\mit\Omega}\times\mathbb R^m\rightarrow\mathbb R^d$ is a Carathéodory function. We obtain general sufficient conditions, under which, for some negligible set $A_\phi$, the integral $I(\phi,\nu_x):=\int_{\mathbb R^m}\phi(x,\lambda)\,d\nu_x(\lambda)$ exists for $x\in{\mit\Omega}\setminus A_\phi$ and $\overline{\phi}(x)=I(\phi,\nu_x)$ on ${\mit\Omega}\setminus A_\phi$, where $\nu=\{\nu_x\}_{x\in{\mit\Omega}}$ is a measurable-dependent family of Radon probability measures on $\mathbb R^m$.
