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The Gibbs measures of a spin system on $Z^d$ with unbounded pair interactions$J_{xy} \sigma (x) \sigma (y)$ are studied. Here $\langle x, y \rangle \in E $,i.e. $x$ and $y$ are neighbors in $Z^d$. The intensities $J_{xy}$ and the spins$\sigma (x) , \sigma (y)$ are arbitrary real. To control their growth weintroduce appropriate sets $J_q\subset R^E$ and $S_p\subset R^{Z^d}$ and provethat for every $J = (J_{xy}) \in J_q$: (a) the set of Gibbs measures $G_p(J)=\{\mu: solves DLR, \mu(S_p)=1\}$ is non-void and weakly compact; (b) each$\mu\inG_p(J)$ obeys an integrability estimate, the same for all $\mu$. Next westudy the case where $J_q$ is equipped with a norm, with the Borel$\sigma$-field $B(J_q)$, and with a complete probability measure $\nu$. We showthat the set-valued map $J \mapsto G_p(J)$ is measurable and hence there existmeasurable selections $J_q \ni J \mapsto \mu(J) \in G_p(J)$, which are randomGibbs measures. We prove that the empirical distributions $N^{-1} \sum_{n=1}^N\pi_{\Delta_n} (\cdot| J, \xi)$, obtained from the local conditional Gibbsmeasures $\pi_{\Delta_n} (\cdot| J, \xi)$ and from exhausting sequences of$\Delta_n \subset Z^d$, have $\nu$-a.s. weak limits as $N\rightarrow +\infty$,which are random Gibbs measures. Similarly, we prove the existence of the$\nu$-a.s. weak limits of the empirical metastates $N^{-1} \sum_{n=1}^N\delta_{\pi_{\Delta_n} (\cdot| J,\xi)}$, which are Aizenman-Wehr metastates.Finally, we prove the existence of the limiting thermodynamic pressure undersome further conditions on $\nu$.

Gibbs measures of disordered lattice systems with unbounded spins