I am confronted with showing the finiteness of the multivariate Laplace transform for a specific measure. Consider a stochastic process $X$ that satisfies $\mathbb{E}[e^{X(t)}]<\infty$. Furthermore consider the finite-dim. distribution of $(X(t_1),...,X(t_m))$ to be $\mathbb{P}^X_{(t_1,...,t_m)}$. Then I want to show that its Laplace transform is defined by \begin{equation} \mathcal{L}\{\mathbb{P}^X_{(t_1,...,t_m)}\}:\mathbb{R}^m\to\mathbb{R}^+, s\mapsto \mathcal{L}\{\mathbb{P}^X_{(t_1,...,t_m)}\}(s)=\int\limits_{\mathbb{R}^m}e^{\langle s,x\rangle}d\mathbb{P}^X_{(t_1,...,t_m)}(x) \end{equation} for $s\in\mathbb{R}^m$ is finite for $s\in S=\{s\mid s_i\in[0,1], \sum_i s_i\leq 1\}$.
I do not seem to understand as much of the solution as I want to understand. I know that, by applying Hölder's inequality we have that $\log\mathcal{L}\{\mathbb{P}^X_{(t_1,...,t_m)}\}$ is a convex function and the set on which the Laplace transform is finite is a convex set. However I don't know why it is precisely the set S. Maybe someone can shed light on this.
Thanks in advance and best regards
Chris
Edit: The original question contained no information on the measure, which was misleading and made the question impossible to answer. Thanks to Kavi Rama Murthy and Robert Israel for pointing it out.