We consider the probability space $ (\mathbb{R},\mathcal{B}(\mathbb{R}),\gamma)$ where $ \gamma$ is the gaussian measure i.e. for all $ A\in\mathcal{B}(\mathbb{R})\,; \gamma(A)=\frac{1}{\sqrt{2\pi}}\int_Ae^{-x^2/2}dx.$

If we take $ \mathcal{S}$ to be the space of $ C^\infty$ functions $ f:\mathbb{R}\to\mathbb{R}$ such that $ f$ and all its derivatives have at most polynomial growth.

Now for such that $ f,$ we look at $ $ \int_{\mathbb{R}}f(x)d\gamma(x).$ $

Does-it follows that $ x$ can be seen as a gaussian random variable ?