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I am looking for a reference, where the conditions for the existence of the n-dimensional Laplace transform are proven, i.e. when the laplace transform

\begin{equation} F(\lambda_1, ..., \lambda_n) = \int_0^\infty dx_1 ... \int_0^\infty dx_n e^{- \sum_{i=1}^n \lambda_i x_i} f(x_1, ..., x_n) \end{equation} exists.

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