While spontaneous symmetry breaking can occur for global symmetries, Elitzur's theorem states that the same is not the case for gauge symmetries; all vacuum expectation values of gauge non-invariant operators are vanishing, even in systems of infinite size. On the lattice this follows from the fact that integrating gauge non-invariant observables over a group measure always yields zero for compact gauge groups. Positivity of the measure and gauge invariance are sufficient to prove the theorem. This is also an explanation for why gauge symmetries are mere redundancies in lattice field theories, where the equations of motion need not define a well-posed problem as they do not need to be solved. Instead, Elitzur's theorem shows that any observable that is not invariant under the symmetry has a vanishing expectation value, making it unobservable and therefore redundant.