# Integral over gaussian measure

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 ?