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lower semicontinuous hull; jointly measurable function; measurable projection theorem; normal integrand
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.
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