Let \((\Omega ,{\mathcal F})\) be a measurable space, \(T\) a locally compact separable metric space, and \(E\) a real separable Banach space. The following theorem is the main result of this paper: Let \(X:\Omega \times T\to 2^E\) be a Carathéodory set-valued mapping, i.e., measurable in \(\omega\) and continuous in \(t\). Then there exists a Carathéodory function \(x:\Omega \times T\to E\) which is a selection of \(\text{{clco}}(X)\). In the proof the authors define \(\hat{X}(\omega )\) as the set of all continuous selections of \(\text{{clco}}(X(\omega ,\cdot ))\), and show that the set-valued function \(\hat{X}\) satisfies assumptions of the Kuratowski and Ryll-Nardzewski measurable selection theorem. The paper is concluded with an application to random fixed points of multivalued random operators.