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We develop the basic theory of ergodic Schr\"odinger operators, which is wellknown for ergodic probability measures, in the case of a base dynamics on aninfinite measure space. This includes the almost sure constancy of the spectrumand the spectral type, the definition and discussion of the density of statesmeasure and the Lyapunov exponent, as well as a version of the Pastur--Ishiitheorem. We also give some counterexamples that demonstrate that some resultsdo not extend from the finite measure case to the infinite measure case. Theseexamples are based on some constructions in infinite ergodic theory that may beof independent interest.

Ergodic Schr\"odinger Operators in the Infinite Measure Setting