Borel-TIS inequality

Statement
Let $$T$$ be a topological space, and let $$\{ f_t \}_{t \in T}$$ be a centred (i.e. mean zero) Gaussian process on $$T$$, with


 * $$\| f \|_T := \sup_{t \in T} | f_t |$$

almost surely finite, and let


 * $$\sigma_T^2 := \sup_{t \in T} \operatorname{E}| f_t |^2.$$

Then $$\operatorname{E}(\| f \|_T)$$ and $$\sigma_T$$ are both finite, and, for each $$u > 0$$,


 * $$\operatorname{P} \big( \| f \|_T > \operatorname{E}(\| f \|_T) + u \big) \leq \exp\left( \frac{- u^2}{2\sigma_T^2} \right). $$