The Young Measure Representation for Weak Cluster Points of Sequences in $M$-spaces of Measurable Functions

Volume 56 / 2008

Hôǹg Thái Nguyêñ, Dariusz P/aczka Bulletin Polish Acad. Sci. Math. 56 (2008), 109-120 MSC: 28A33, 46E27, 46E30, 46E40, 47H30. DOI: 10.4064/ba56-2-3


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 $\mathbb R^m$.


  • Hôǹg Thái NguyêñInstitute of Mathematics
    Szczecin University
    Wielkopolska 15
    70-451 Szczecin, Poland
  • Dariusz P/aczkaInstitute of Mathematics
    Szczecin University of Technology
    Al. Piastów 48
    70-311 Szczecin, Poland

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