Without apparatus can we say that the system is measured(decohered) by the environment? "Einselection" and "tridecompositional uniqueness theorem" seem to resolve the preferred basis problem. But the premise is that there are three parts in discussion.(system, apparatus, environment)
However, it seems that in many situations we don't have the role of apparatus, and thus there are just system and environment. For instance, we usually say that the system is monitored by its environment and thus in a state with determinate classical physical value. 
In these situations, in which there are just two parts(system and environment) and the Schmidt form of the total system is not unique, can we say  that the system is measured(decohered) by the environment? 
 A: At least to me, it is unclear what it means to be "measured by the environment". As far as decoherence is concerned the situation is however quite clear.
Already the original "einselection" framework of Zurek is applicable to bipartite system/environment scenarios.
Let $(| p\rangle)_p$ be a "pointer basis" for the system. Then any Hamiltonian of the form 
$$
H = \sum_p |p\rangle\langle p| \otimes H^{(p)} ,
$$
with $(H^{(p)})_p$ being some Hamiltonians of the environment, leads to a time evolution that, if the joint system starts in a product state, leaves the diagonal elements of the density matrix of the system invariant and "suppresses" the off-diagonal elements, in the sense that they are during the whole evolution never larger than they were initially and usually for most times very small.
A similar phenomenon can also be shown for more generic Hamiltonians under the assumption that the coupling between the system and the environment is sufficiently weak (see for example http://arxiv.org/abs/0908.2921 and the references therein). 
A: First a word about "measurement". Measurement for the purposes of decoherence is just an interaction such that information is initially present in one system becomes present in more than one system.  Calling one of the systems involved in an interaction a measuring system just means that the "measurement" is arranged in such a way that it is easy for us to make a record of the result, which is irrelevant as far as the system being decohered is concerned. So the relevant issue is just whether there is more than one system that can be treated as independent that has "measured" information about the system. See http://arxiv.org/abs/1212.3245.
