Definition 2

Let $E$ be a locally convex space. For every positive quadratic form $Q$ on $E'$, the promeasure on $E$ whose Fourier transform is equal to $e^{-Q/2}$ is called the Gaussian promeasure on $E$ with variance $Q$, and is denoted $\Gamma_Q$. A promeasure $\mu$ on $E$ is said to be Gaussian if there exists a positive quadratic form $Q$ on $E'$ such that $\mu = \Gamma_Q$.

By an abuse of language, a bounded measure $\mu$ on $E$ will be said to be Gaussian with variance $Q$ if the associated promeasure $\widetilde{\mu}$ is equal to $\Gamma_Q$.