E\in I:\, \text{ $\ball_L(x)$ and $\ball_L(y)$ are $(E,m)$-S}}$, with $\pr{\CN_L}\le f(L)$, by assumption. Further, $$\esm{ \mu_{x,y}(\phi) } = \esm{ \operatorname{\mathbf{1}}_{\CN_L}\mu_{x,y}(\phi) }
   + \esm{ \operatorname{\mathbf{1}}_{\CN_L}\mu_{x,y}(\phi) }  \le f(L) + 2CL^d {\mathrm{e}}^{-mL}.
\qedhere$$

Strong dynamical localization on the entire lattice {#ssec:SimpleDL.lattice}
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Here we follow the same strategy as in earlier works by Aizenman et al. [@A94; @ASFH01].

\[thm:DL.lattice.IID\] Consider the Hamiltonian $H(\om)$ of the form with random potential satisfying the assumption , and fix an interval $I\subset\DR$. There exists a $L_0^* < \infty$ with the following property: if the conditions and hold for some $L_0 \ge L_0^*$, then for all $x,y\in\DZ^d$, $x\ne y$, and some $c, a>0$, \[eq:DL.bound.any.s.lattice.IID\] x-y\^[-a \^c x-y ]{}.

It suffices to use an argument given earlier in [@ASFH01]. For any ball $\ball$ and any points $x,y\in\ball$ introduce a spectral measure $\mu^{x,y}_{\ball,\om}$ uniquely defined, for any bounded Borel function $f$, by $$\int\, f(\lambda)\, d\mubxy(\lambda)
= \bra{\delta_x} f(H_\ball(\omega)) \BP_I(H_{\ball}(\omega)) \ket{\delta_y},$$ and similar spectral measures $\muxy (\equiv \mu^{x,y}_{\DZ^d,\om})$ for the operator $H(\om)$ on the entire lattice. Then $\mukxy$ converge vaguely to $\muxy$ as $k\to\infty$, so that by virtue of Fatou lemma on convergent measures, for any measurable set $\CE\subset\DR$ $$\esm{ |\muxy| (\CE) } \le \liminf_{k\to\infty} \; \esm{ |\mukxy| (\CE) }.$$ 