  We consider one-dimensional translationally invariant quantum spin (or
fermionic) lattices and prove a Mazur-type inequality bounding the
time-averaged thermodynamic limit of a finite-temperature expectation of a
spatio-temporal autocorrelation function of a local observable in terms of
quasi-local conservation laws with open boundary conditions. Namely, the
commutator between the Hamiltonian and the conservation law of a finite chain
may result in boundary terms only. No reference to techniques used in Suzuki's
proof of Mazur bound is made (which strictly applies only to finite-size
systems with exact conservation laws), but Lieb-Robinson bounds and exponential
clustering theorems of quasi-local C^* quantum spin algebras are invoked
instead. Our result has an important application in the transport theory of
quantum spin chains, in particular it provides rigorous non-trivial examples of
positive finite-temperature spin Drude weight in the anisotropic Heisenberg XXZ
spin 1/2 chain [Phys. Rev. Lett. 106, 217206 (2011)].
