The measurements of the potential distributions in the boundary layer near meshes with different mesh spacing were conducted in weakly collisional plasmas using a fine-structured emissive probe and the results of the sheath thickness and electric field at the sheath-presheath edge were compared with theoretical models of collisional presheath and collisionless sheath. It was shown that, because the meshes are partially transparent to ions, the sheath is thinner and the electric field is stronger for the mesh of higher transmissivity, owing to the increased ion density in the sheath contributed from the ions transmitted from the other side of the mesh. However, the potential profiles in the presheath remain almost the same for different meshes except for the shift of the sheath-presheath edge. The thickness of the sheath decreases while the electric field at the edge increases with the increase of the neutral gas pressure. Furthermore, depending on the pressure, the measured electric fields at the edge are close to that from the models of a transition region.