Let $T$ be an codimension one rectifiable mass-minimizing current. If $\partial T\llcorner B_R(a)=0$, then it is possible to write $T$ as an infinite sum of oriented boundaries: $T\llcorner B_R(a)=\sum_{j=-\infty}^\infty \partial [[E_j]]$ s.t. the masses are additive $\mathbf{M}(T\llcorner B_R(a)) = \sum \mathbf{M}( \partial [[E_j]])$. Furthermore every current $\partial[[E_j]]$ itself is mass-minimizing.
I understand this decomposition theorem. My question is the following:
Frank Morgan writes in his book (Ch 10): Since $\mathbf{M}(T\llcorner B_R(a)) < \infty$, it follows from monotinicity that $\text{spt} \partial [[E_j]]$ intersects $B_{R/2}(a)$ for only finitely many $i$.
He quotes the monotonicity facts: $\mathbf{M}(T\llcorner B_R(a)) \ge \Theta(T,a) \omega_nr^n$ and $\Theta(T,a)\ge 1$.
I do not understand why one has to consider a smaller neighbourhood $B_{R/2}(a)$ ??? I think, if the density of every mass-minimizing current $\partial [[E_j]]$ is greater than $1$, then it follows directly that it is impossible to have infinitely many, otherwise the mass of $T$ would be $\infty$...?
Thanks for helping me.
