The following picture is lemma 4.23 in *Lectures on Coarse Geometry* by John Roe:

[![enter image description here][1]][1]

I guess the $E_i$ in the centered formula is $X_i$. Does Roe mean that $X_j\cap \mathrm{Supp}(u)=\emptyset $ implies  $\lambda(E_j)u=0$? But I can't work out. Can someone help me? (If more details such as  definitions are needed, let me know.)

  [1]: https://i.sstatic.net/32spt.png