The Integral Equation for Stokes flow

I am trying to understand and rederive why "The Integral Equations for Stokes Flow" is convoluted over just the surfaces of the rigid boundaries, and not the entire fluid volume.

$$u_i(boldsymbol x)=u_i^{infty}(boldsymbol x)-dfrac{1}{8pimu}sum_{alpha=1}^Nint_{S_alpha}J_{ij}(boldsymbol x – boldsymbol y )f_j(boldsymbol y ) , dS_y$$

where $J_{ij}$ is the stokeslet, $N$ is the number of particles and $f_j(boldsymbol y)=sigma_{ij}(boldsymbol y) cdot n_k(boldsymbol y)$. My current understanding is that you convolve over the entire domain when utilizing a greens function, so maybe you use Gauss’s Theorem? I am looking for any insight as to why this is and how to show it step by step.