On a generalization of stochastic integral

Let $\xi$ be a Gaussian random variable in a separable Hilbert space $H$ and $L$ be the space of random variables $\eta$ in $H$ with $\mathbf M|\eta|^2<\infty$. In the paper, the integral $\langle\eta,\xi\rangle$ is introduced and its properties are investigated. If $H$ is the space of those functions $f(x)$ on a measurable space $(X,\mathfrak B)$ for which
$$ \int f^2(x)m(dx)<\infty $$
and $\mu$ is a Gaussian measure on $\mathfrak B$ with
$$ \mathbf M\mu(A)\mu(B)=m(A\cap B), $$
then the integral
$$ \langle\eta(\,\cdot\,),\mu\rangle=\int\eta(x)\mu(dx). $$