Cylinder set measure

Remarks

A cylinder set is a set $T^{-1}(B)$ with $T \in \mathcal{T}_n$ and $B \subseteq \R^n$ Borel. Condition 2 makes the rule $\mu(T^{-1}(B)) = \mu_T(B)$ well defined, and $\mu$ is a finitely additive measure on the algebra of cylinder sets.

Example: a Borel probability measure $\nu$ on $E$ gives the cylinder set measure $\mu_T = T_* \nu$. Condition 2 holds since $\pi_*(S_* \nu) = (\pi \circ S)_* \nu$.

Example: let $E = H$ be a real Hilbert space and let $\mu_T$ be the centered normal law on $\R^n$ with covariance matrix $T T^*$. Condition 2 holds, because $\pi_*$ of the centered normal law with covariance $S S^*$ is centered normal with covariance $\pi S S^* \pi^*$. This is the canonical Gaussian cylinder set measure on $H$.

Non-example: on $E = \R^2$, let every $\mu_T$ with $T \in \mathcal{T}_1$ be the standard normal distribution $N(0, 1)$. For $T(x) = x_1$ and $T'(x) = 2 x_1$, one has $T' = \pi \circ T$ with $\pi(t) = 2t$, and $\pi_* N(0, 1) = N(0, 4) \ne N(0, 1)$, so condition 2 fails.