Weakly Gibbsian measures for lattice spin systems

We present a weaker notion of Gibbs measures by requiring that only an almost everywhere absolutely summable potential is given. This has recently appeared in the context ''Gibbsian versus non-Gibbsian measures''. We give a first exploration of the main Features of this weaker notion. We concentrate on the questions of where do weakly Gibbsian measures appear and what remains of their thermodynamic description.