I am reading about $\gamma$-radonifying operators, and came across 2 similar definitions. And I hoped to understand why they are equivalent.
Let $H$ be a seperable real Hilbertspace, $E$ banach space.
(1). A bounded linear operator $T:H\to E $ is said to be $\gamma$-radonifying if $T_{\#}\mu$ is a Radon measure for every Gaussian cylindrical measure $\mu$ on $H$.
(2). $T:H\to E$ (bounded linear) is $\gamma$-radonifying if $T_{\#}\gamma$ is a Radon-measure for the standard cylindrical gaussian $\gamma$.
the image measure $T_{\#}\mu$ is the measure defined by $T_{\#}\mu(A) = \mu(T^{-1}(A))$. Anyone familiar with the Theory of Gaussian random variables on Banach-spaces, Gaussian measures, and radonifying operators? Any ideas on why these definitions are equivalent are very welcome.