  We consider self-affine tiling substitutions in Euclidean space and the
corresponding tiling dynamical systems. It is well-known that in the primitive
case the dynamical system is uniquely ergodic. We investigate invariant
measures when the substitution is not primitive, and the tiling dynamical
system is non-minimal. We prove that all ergodic invariant probability measures
are supported on minimal components, but there are other natural ergodic
invariant measures, which are infinite. Under some mild assumptions, we
completely characterize $\sigma$-finite invariant measures which are positive
and finite on a cylinder set. A key step is to establish recognizability of
non-periodic tilings in our setting. Examples include the "integer Sierpi\'nski
gasket and carpet" tilings. For such tilings the only invariant probability
measure is supported on trivial periodic tilings, but there is a fully
supported $\sigma$-finite invariant measure, which is locally finite and unique
up to scaling.
