The Young Measure Representation for Weak Cluster Points of Sequences in -spaces of Measurable Functions
Tom 56 / 2008
Streszczenie
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.
