Standard Gaussian probability space

The standard Gaussian probability space in dimension $d\geq1$ is $(\mathbb R^d,\Sigma,\gamma_d)$ , where $\Sigma$ is the completed Borel sigma-algebra and

\gamma_d(E)=(2\pi)^{-d/2}\int_E e^{-\|x\|^2/2}\,dx.

It is the product of $d$ standard normal laws. Its mean is zero and its covariance matrix is $I_d$ .

Observable spaces

Although $\mathbb R^d$ has finite dimension, $L^2(\gamma_d)$ and $L^\infty(\gamma_d)$ are infinite dimensional. The probability space is atomless and standard. Integration defines a state on the bounded function algebra.

A useful nonlinear observable

For $r(x)=\|x\|^2-d$ , independence and the standard normal moments give $\int r\,d\gamma_d=0$ and $\|r\|_2^2=2d$ . This nonzero centered observable is fixed by every orthogonal transformation.