Assume I have a collection of real-valued measurable functions $(X_i)_{i \in I}$ on the measurable space $(\Omega,\mathcal{F})$. Let $N:\Omega \rightarrow 2^\Omega$ such that for every $\omega \in \Omega$, the value $|N(\omega)| < \infty$, where $|\cdot|$ now denotes the cardinality of a set. If I look at the function $S:\Omega \rightarrow \mathbb{R}$ defined by $$S(\omega):= \sum_{i \in N(\omega)}X_i(\omega)$$ for $\omega \in \Omega$, can I somehow deduce that $S$ is measurable as a function from $(\Omega,\mathcal{F}) \rightarrow (\mathbb{R},\mathcal{B}(\mathbb{R}))$ ?
Thanks a lot in advance!
Not without some sort of measurability condition on $N$. Exactly what condition that would be I don't know, but some condition is needed:
Say $X_0=0$, $X_1=1$. Say $E\subset\Omega$ is not measurable, and define $$N(\omega)=\begin{cases}\{0\},&(\omega\in E), \\\{0,1\},&(\omega\notin E).\end{cases}$$