Filippov implicit function theorem for quasi-Carathéodory functions (Q1380363)

Let \((T,{\mathcal A})\) be a measurable space, \(X\), \(Y\) topological spaces, \(f: T\times X\to Y\) a function, \(\Gamma: T\to 2^X\) a multifunction, and \(g: T\to Y\) a function such that \(g(t)\in f(\{t\}\times \Gamma(t))\), \(t\in T\). The authors give sufficient conditions for the existence of a measurable selector \(\gamma\) of \(\Gamma\) such that \(g(t)= f(t,\gamma(t))\), \(t\in T\). In the main result of the paper, Theorem 1, they assume that \(f\) is measurable in \(t\) and quasicontinuous in \(x\), \(\Gamma\) is measurable and compact-valued, and \(g\) is measurable and such that for each \(t\in T\), \(g(t)\) is a closed value of \(f(t,\cdot)\), i.e., for each sequence \((x_n)\) the following implication holds: \(x_n\to x\) and \(f(t,x_n)\to g(t)\) implies \(f(t, x)= g(t)\). The proof is based on the Kuratowski and Ryll-Nardzewski selection theorem.