Corollary 2

— Let $Q$ be a nuclear positive quadratic form on the Hilbert space $E$. The Gaussian promeasure $\Gamma_Q$ on $E$ with variance $Q$ is a measure on $E_s$.

The Fourier transform $\Phi$ of $\Gamma_Q$ is equal to $e^{-Q/2}$. Now, $e^t \geq 1 + t$ for every real number $t$, whence $\Phi(0) - \Re \Phi \leq Q/2$. The condition b) of Theorem 3 is therefore verified and $\Gamma_Q$ is a measure on $E_s$.