# Article

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Keywords:
lower semicontinuous hull; jointly measurable function; measurable projection theorem; normal integrand
Summary:
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.
References:
[1] Balder E.J.: A general approach to lower semicontinuity and lower closure in optimal control theory. SIAM J. Control Optim. 22 (1984), 570–598. DOI 10.1137/0322035 | MR 0747970 | Zbl 0549.49005
[2] Balder E.J.: Generalized equilibrium results for games with incomplete information. Math. Oper. Res. 13 (1988), 265–276. DOI 10.1287/moor.13.2.265 | MR 0942618 | Zbl 0658.90104
[3] Balder E.J.: On ws-convergence of product measures. Math. Oper. Res. 26 (2001), 494–518. DOI 10.1287/moor.26.3.494.10581 | MR 1849882 | Zbl 1073.60500
[4] Carbonell-Nicolau O., McLean R.P.: On the existence of Nash equilibrium in Bayesian games. mimeograph, 2013.
[5] Cohn D.L.: Measure Theory. Second edition, Birkhäuser/Springer, New York, 2013. MR 3098996 | Zbl 0860.28001
[6] Sainte-Beuve M.-F.: On the extension of von Neumann-Aumann's theorem. J. Functional Analysis 17 (1974), 112–129. DOI 10.1016/0022-1236(74)90008-1 | MR 0374364 | Zbl 0286.28005

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