Semicontinuous integrands as jointly measurable maps

Carbonell-Nicolau, Oriol

Commentationes Mathematicae Universitatis Carolinae, Tome 55 (2014) no. 2, pp. 189-193.

Résumé

Suppose that $(X,\mathcal A)$ is a measurable space and $Y$ is a metrizable, Souslin space. Let $\mathcal A^u$ denote the universal completion of $\mathcal A$. For $x\in X$, let $\underline f(x,\cdot)$ be the lower semicontinuous hull of $f(x,\cdot)$. If $f:X\times Y\rightarrow\overline{\mathbb R}$ is $(\mathcal A^u\otimes\mathcal B(Y),\mathcal B(\overline{\mathbb R}))$-measurable, then $\underline f$ is $(\mathcal A^u\otimes\mathcal B(Y),\mathcal B(\overline{\mathbb R}))$-measurable.

Détail

Classification : 28A20, 54C30
Keywords: lower semicontinuous hull; jointly measurable function; measurable projection theorem; normal integrand