Given a probability space $(\Omega,\mathcal{A},\mathbb{P})$, we have a random field $\{X_t\}_{t \in T}$, $T\subset S_1\times S_2$, for a measurable space $(S_1 \times S_2,\mathcal{A}_1\times\mathcal{A}_2)$, i.e. $\forall t \in T$ $X_t:\Omega \to \mathbb{R}$, $\omega \mapsto X_t(\omega)$ is $\mathcal{A}/\mathcal{B}(\mathbb{R})$ measurable.

Now define $\mu:\Omega\times S_1\times S_2\times \mathbb{R} \to \mathbb{R}$ by \begin{equation} \mu(\omega\times s1,A_2,B)=\sum_{s_2 \in A_2}\delta_{X_{(s1,s2)}(\omega)}(B), \ B \in \mathcal{B}(\mathbb{R}), \ A_2 \in \mathcal{A}_2. \end{equation}

Is $\mu$ a random measure, i.e.

1) $\mu(\omega\times s1,\cdot,\cdot)$ is a measure on $S_2\times\mathbb{R}$,

2) $\mu(\cdot,A_2,B)$ is $\mathcal{A}\times\mathcal{A}_1/\mathcal{B}(\mathbb{R})$ measurable?

I think that $\mu(\omega\times s1,\cdot,\cdot)$ should be a measure. However, I have no idea how I can ensure measurability, i.e. condition 2). Maybe someone has also an idea for a similar construction of $\mu$.


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