Multidimensional Gaussian white noise definition

Question

In one of the papers I came across the following:

Let ${W(dt, du): t, u ≥ 0}$ denote a Gaussian white noise with density measure $dtdu$ on $(0,\infty)^2$.

Now, if I understand this correctly, $W$ can be defined as a random measure satisfying

1. $W(A) \sim \mathcal{N}(0, \lambda (A))$, where $\lambda (A)$ is the Lebesgue measure of $A \subset (0,\infty)^2$,
2. $W(A_1), ..., W(A_n) $ are independent if $A_1,... A_n$ have pair-wise empty intersections.

Where can I find this definition and basic properties of $W$ and the integral with respect to $W$? I am looking for a book or an article.