 Tail event for a vanishing positive path-weight limit

Delete the first $m$ environment factors from the path-weight expectation and call the result $Z_T^{(m)}$. Pointwise, $(1-\epsilon {)}^m{Z}_T^{(m)}\le Z_T\le (1+\epsilon {)}^m{Z}_T^{(m)}$. Thus $\{{\mathrm{liminf}}_T{Z}_T=0\}$ equals $\{{\mathrm{liminf}}_T{Z}_T^{(m)}=0\}$ and belongs to every future-time environment sigma-algebra. The Kolmogorov zero-one law makes its probability zero or one. If the nonnegative martingale limit has mean one, the probability cannot be one, so its limit is strictly positive almost surely. The pointwise comparison establishes a genuine tail event, rather than merely invariance outside an unspecified exceptional set.