Characterization of Poisson random measure in terms of Laplace transform

Question

Let $(E,\mathcal E)$ be a measurable space and $\mu$ be a measure on $(E,\mathcal E)$.

A random measure $\pi$ on $(E,\mathcal E)$ is called Poisson with intensity $\mu$ if

1. $\pi(B)\sim\operatorname{Poi}(\mu(B))$ (Poisson distribution with intensity $\mu(B)$) for all $B\in\mathcal B(E)$
2. $\pi(B_1),\ldots,\pi(B_k)$ are independent for all $k\in\mathbb N$ and disjoint $B_1,\ldots,B_k\in\mathcal E$.

It's easy to show that (1.) and (2.) together are equivalent to

3. $\operatorname E\left[e^{-\pi f}\right]=e^{-\mu\left(1-e^{-f}\right)}$ for all $\mathcal E$-measurable $f:E\to[0,\infty)$ (, where I wrote $\kappa f:=\int f\:{\rm d}\kappa$).

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Now if we replace the space $(E,\mathcal E)$ by the product space $([0,\infty)\times E,\mathcal B([0,\infty)\otimes\mathcal E)$ and $(\mathcal F_t)_{t\ge0}$ is a filtration on $(\Omega,\mathcal A)$, we may say that a Poisson random measure on $([0,\infty)\times E,\mathcal B([0,\infty)\otimes\mathcal E)$ with intensity $\left.\lambda\right|_{[0,\:\infty)}\otimes\mu$ (, where $\lambda$ is the Lebesgue measure on $\mathcal B(\mathbb R)$,) is called $\mathcal F$-Poisson if

4. $\pi(B)$ is $\mathcal F_t$-measurable for all $B\in\mathcal B([0,t])\otimes\mathcal E$; and
5. $\left.\pi\right|_{(t,\:\infty)\times E}$ (the trace of $\pi$ on $(t,\infty)\times E$) is independent of $\mathcal F_t$

for all $t\ge0$.

Question: Is there a similar characterization of $\pi$ being $\mathcal F$-Poisson in terms of the Laplace transform as before? I would be thankful for any answer or reference.