Measurability of the argmin 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.
 A: 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}$.
