Gaussian measures on Orlicz spaces and abstract Wiener spaces (Q1081947)

Let (X,\(\tau)\) be a measurable linear space and \(\mu\) be a probability measure on \(\tau\). A measurable functional \(f: X\to {\mathbb{R}}\) is called \(\mu\)-quasi-additive if \(f(x\pm y)=f(x)\pm f(y)\mu\times \mu\) a.e.. The set of all \(\mu\)-quasi-additive functionals is denoted by \(X_{\mu}^{*}\). The space \(X_{\mu}^{*}\) is studied when \(\mu\) is a Gaussian measure. If X is Orlicz space with Gaussian measure \(\mu\), then the inclusion map \(j: X_{\mu}^{*}\to X\) is defined. This map has the property that if X is a separable Banach space and f is a continuous linear functional, then \[ jf=\int_{X}xf(x)d\mu (x). \] The image \(j(X_{\mu}^{*})\) is treated as RKHS of \(\mu\) and thus \(\mu\) is considered as abstract Wiener measure. As a consequence, the LIL for Gaussian random elements in Orlicz space is formulated.