Typically, the ETH is postulated to hold for "few-body operators," observables which involve only a small number of particles. Examples of this would include the occupation of a given momentum in a gas of particles, or the occupation of a particular site in a lattice system of particles. Notice that while the ETH is typically applied to "simple" few-body operators such as these, these observables need not be local in space - the momentum number operator in the above example does not represent a local quantity.