Suppose $ \Gamma$ is a compact space and I have a stochastic process $ ( X ( \gamma) , \gamma \in \Gamma ) $ on a probability space $ ( \Omega , \mathscr{A} , \mathbb{P})$ such that $ \gamma \mapsto X ( \gamma) $ is continuous $ \mathbb{P}$-a.e. Then, $\arg\!\min_{ \gamma \in \Gamma} X ( \gamma) $ exist $ \mathbb{P}$-a.e. My question is, whether there is a measurable function $ \widehat{ \gamma} : ( \Omega , \mathscr{A} , \mathbb{P}) \to ( \Gamma, \mathscr{B}_{ \Gamma}) $ with $ \widehat{ \gamma} = \arg\!\min_{ \gamma \in \Gamma} X ( \gamma) $ $ \mathbb{P} $-a.e.,where $ \mathscr{B}_{ \Gamma}$ is the borel $ \sigma$-Algebra on $ \Gamma$.
Any help would be appreciated.
I have found a solution for $ \Gamma \subset \mathbb{R}^{ k}$ I would like to share here for anyone else with the same problem. The reasoning follows Witting, Mathematische Statistik II, Satz 6.7. Consider the case $ \Gamma = \left[ 0 , 1 \right] $. Since $ \Gamma$ is compact and $ \gamma \mapsto X( \gamma) $ is continuous $ \mathbb{P}_{ \theta_{ 0}}^{ n}$-a.e., $ \arg\!\min_{ \gamma \in \Gamma} X ( \gamma) \ne \emptyset $ $ \mathbb{P}$-a.e. and let $ N \in \mathscr{A}$ be the corresponding $ \mathbb{P}$-null-set on which this is not the case. Since $ \Gamma$ is compact, $ \exists ( \gamma_{ j})_{ j \in \mathbb{N}} $ dense in $ \Gamma$ and $ Y : = \mathbf{1}_{ N^{ c} } \min_{ \gamma \in \Gamma} X ( \gamma) = \mathbf{1}_{ N^{ c}} \inf_{ j \in \mathbb{N}} X ( \gamma_{ j}) $ is a well defined random variable. Since $ \gamma \mapsto X ( \gamma) $ is continuous, $ \arg\!\min_{ \gamma \in \Gamma} X ( \gamma) = \left\{ \gamma \in \Gamma : X ( \gamma) = Y \right\} $ is compact on $ N^{ c}$. Therefore, $\widehat{ \gamma} : = \mathbf{1}_{ N^{ c}} \sup_{ } \arg\!\min_{ \gamma \in \Gamma} X ( \gamma) $ is well defined as a mapping. Finally, $ \widehat{ \gamma}$ is measurable since for any $ \alpha \in \mathbb{R}$, the continuity of $ X $ on $ N^{ c}$ implies that \begin{align*} \left\{ \widehat{ \gamma} < \alpha \right\} & = N \cap \left\{ 0 < \alpha \right\} \, \cup \, N^{ c} \cap \left\{ \forall \gamma \in \Gamma : X ( \gamma) = Y\Rightarrow \gamma < \alpha \right\} \\ & = N \cap \left\{ 0 < \alpha \right\} \, \cup \, N^{ c} \cap\left\{ \forall \gamma \ge \alpha : X ( \gamma) > Y \right\} \\ & = N \cap \left\{ 0 < \alpha \right\} \, \cup \, N^{ c} \cap \bigcap_{ \gamma_{ j} > \alpha }^{ } \left\{ X ( \gamma_{ j} ) > Y \right\} \cap \left\{ X ( \alpha) > Y \right\} \in \mathscr{A}. \end{align*} For any other compact set in $ \mathbb{R}^{ k}$, the result can be obtained with exactly the same reasoning, if we impose a lexicographical ordering on $ \mathbb{R}^{ k}$.