In decoherence theory, we describe the decoherence of a system $S$ by the mean of an interaction with an environment. In short, if I consider: $|\psi_S \rangle=a|0_S\rangle+b|1_S\rangle$, the system will interact with the environment and I will have:
$$\left( a|0_S\rangle + b|1_S\rangle \right) |0_E\rangle \rightarrow a|0_S0_E(t)\rangle +b|1_S1_E(t)\rangle$$
Tracing out the environment, we have, with $r(t)=\langle 1_E(t) | 0_E(t) \rangle$
$$\rho_{S}=|a|^2 |0\rangle \langle 0| + |b|^2 |1 \rangle \langle 1 | +ab^* r(t) |0\rangle \langle 1| + a^*b r^*(t) |1\rangle \langle 0 |$$
On a short timescale, we will have $r(t) \approx 0$, the final state of the system will be a classical mixture. This is usually how we describe decoherence process.
When we talk about a measurement process, replacing $E$ by an apparatus $A$ leaves an un-satisfactory explanation. Indeed if my environment played the role of an apparatus, I would have for the global state, assuming the process completly killed the coherence of the system, and modelling for simplicity the apparatus as a two level system:
$$|\psi_{SA}\rangle= a|0_S0_A\rangle + b|1_S1_A\rangle$$
In the particular case $a=b=1$, this state is also $|++\rangle +|--\rangle$. The measurement is then ill-defined (did we measure $\sigma_x$, or $\sigma_z$: it looks the same in those two basis !). I call it the "what did we measure" problem in what follows.
A way to solve this issue is then to use a third system (the environment) that will remove such ambiguity. Basically at the end we will have a $SAE$ state: $a|0_S0_A0_E\rangle + b|1_S1_A1_E\rangle$ which because of the third system removes all ambiguity (I don't go in the details but this is a result from tri-orthogonal decomposition theorem).
My question is:
I totally understand the "what did we measure" problem with the apparatus. But then it should also be a conceptual problem for a "natural" decoherence (i.e system interacting with environment). However in all environmental decoherence study I looked we always assume system+environment as the total system of study. We do not include some extra part to remove this issue.
Is this ambiguity implicitly removed by the fact the environment $E$ is actually composed of many sub-systems. Basically: $H_E = H_{E_1} \otimes H_{E_2}$, and the state is actually something like:
$$|\psi_{SE_1E_2}\rangle= a|0_S0_{E_1}0_{E_2}\rangle + b|1_S 1_{E_1}1_{E_2}\rangle$$
And I can then interpret the decoherence as a measurement process as well (the part $E_1$ of the environment measured my system).
Would you agree that it is precisely the fact we can "factorise" the environment into two sub-environment that explains the fact we don't have a "what did we measure" problem with the environment ?
