For every $\delta>0$, there exists $r_{0}({\rm v},\delta),N({\rm v},\delta)>0$ with the following properties. Let $M^{4}$ satisfy $|{\rm Ric}_{M^{4}_{j}}|\leq 3\delta$, ${\rm Vol}(B_{1}(p))>{\rm v}>0$. Then there exists points $\{x_{j}\}_{1}^{N}$ with $N\leq N({\rm v},\delta)$, and scales $\alpha_{j}\in\mathds{N}$ with $r_{j}\equiv r_{\alpha_{j}}>r_{0}$, such that

1. (1)

   $T^{\delta}_{\alpha_{j}}(x_{j})=0$,

2. (2)

   If $x\in B_{1}(p)\setminus\bigcup_{j}B_{r_{j}}(x_{j})$ then $r_{h}(x)>r_{0}$,

3. (3)

   If $\beta_{j}\in\mathds{N}$ denotes the largest integer such that $\mathcal{V}^{\delta}_{r_{\beta_{j}}/4}(x_{j})\geq\ln|\Gamma_{j}|-\delta$, then for every $x\in B_{2r_{\beta_{j}}}(x_{j})$ we have

   $\displaystyle\mathcal{V}^{\delta}_{r_{\beta_{j}}/8}(x)<\ln|\Gamma_{j}|-\delta\,.$

   (8.49)