In page 6 from the notes "An introduction to SPDE's" from Walsh, one reads:
The Brownian Sheet, is defined as follows:
The notation $((u,v), (s,t)]$ is a bit unclear also to me, and maybe it is the cause of my confusion.
In $\Bbb{R}^2$, $(0, (s,t)] = (0,s] \times (0,t]$ one might extend this definition to include more base point other than 0. In this spirit one can define for $u \leq s$, $v \leq t$ $$((u,v),(s,t)] = (u,s] \times (v,t].$$
That using this definition one then notes that $$(u,s] \times (v,t] = (0,s] \times (0,t] \setminus ((0,s] \times (0,v] \setminus \bigg( (0,u] \times (0,t] \setminus (0,u] \times (0,v])\bigg)$$
Here is a picture:
The above translates as
$$ 1 = 1234 - 34 - (24 - 4) = 1234 -23 -4 $$
One should then have
$$W((u,s] \times (v,t]) = W_{st}- W_{sv} - (W_{ut} - W_{uv}) = W_{st}- W_{sv} - W_{ut} + W_{uv} $$
But that is not what is written. Is this a typo?