Why should multiple versions of a weak measurement preclude it from being a measurement of intrinsic properties of some system I've been trying to understand Stephen Parrott's criticisms of weak measurement, outlined most concisely here: http://arxiv.org/abs/0909.0295 
One of his major criticisms is that weak measurement is not unique. I'll try to give a brief outline of his argument here:

We consider a system of interest S, and a system of measurement M, each with an associated hilbert space $\mathbb{H}_S$ and $\mathbb{H}_M$. We start with a separable state state $|s\rangle\otimes |m\rangle \in \mathbb{H}_S\otimes\mathbb{H}_M$ and let it evolve for some short time $\epsilon$ under the action of some hamiltonian $H=A\otimes G$. The resulting state is $|r(\epsilon)\rangle=e^{-i\epsilon H}|s\rangle\otimes |m\rangle=|s\rangle\otimes |m\rangle-i\epsilon A|s\rangle\otimes G|m\rangle+ O(\epsilon^2)$. 
If we choose B and G such that $\langle m|B|m\rangle=0$ and $Im(\langle m|BG|m \rangle)=1/2$ then one can calculate that $\lim_{\epsilon ->0}[ \frac{\langle r(\epsilon)| 1 \otimes B |r(\epsilon)\rangle}{\epsilon}] =\langle s|A|s\rangle$.
We then consider a scenario where one repeatedly performs this measurement $\langle r(\epsilon)|1\otimes B|r(\epsilon)\rangle$ followed by a projective postselection measurement $P_f \otimes 1= |f\rangle \langle f| \otimes 1$, throwing away the data from any experiment for which our postselection measurement gives 0. If we take $\epsilon$ to be very small, this procedure can be described by  $NE(B|f)=\lim_{\epsilon ->0}[ \frac{\langle r(\epsilon)| P_f \otimes B |r(\epsilon)\rangle}{\epsilon \langle r(\epsilon)| P_f \otimes 1 |r(\epsilon)\rangle}] =Re[\frac{\langle f|A|s\rangle}{\langle f| s\rangle}]$, an expression which Parrott claims the weak measurement literature identifies as the conditional expectation of A in state $|s\rangle$ given succesful postselection to state $|f\rangle$
Parrott then considers a scenario where $\langle m |BG |m \rangle =\rho +1/2 i$, which satisfies our original constraint when $\rho$ is real, but when inserted into the above procedure yields $NE(B|f)=Re[\frac{\langle f|A|s\rangle}{\langle f| s\rangle}]+2\rho Im[\frac{\langle f|A|s\rangle}{\langle f| s\rangle}]$. From this Parrott deduces that the weak measurement is not unique, by virtue of the variability of $\rho$, and that this casts doubt on the weak measurement actually measuring some intrinsic property of system S.

So I suppose my confusion lies in trying to understand what is preventing a proponent of weak measurement from simply adding an additional constraint on$BG$, namely that  $Re[\langle m |BG|m\rangle]=0$, this would mean $\rho=0$. Then there would be a unique definiton of weak measurement. Is it not the case that the mathematical construction of  the weak measurement is the way it is simply because it results in a convenient expression for $Re[\frac{\langle f|A|s\rangle}{\langle f| s\rangle}]$, and is not this most conveniently expressed when $\rho=0$? 
Even more to the point, how is it necessarily the case that because there is more than one way to make your measurement this implies your measurement is not of some intrinsic property of the system? Parrott gives an example of two voltmeters which give different readings, he correctly states that if one expects them to read the same voltage, and they don't, then they must be making measurements of different things, maybe one or both isn't even measuring voltage. This analogy doesn't seem apt in this scenario, since it seems to me that $\rho$ is akin to some kind of calibration, and so in the two scenarios we would expect them to read different things. 
I should add that I'm aware of the controversial status of weak measurement, and I hold no allegiance to either camp. I merely wish to more thoroughly understand Parrott's arguments, as I suspect I must be missing something fundamental.
Thanks!
 A: I think that Stephen Parrott's criticism is formulated in a so messy way that it is difficult to succinctly summarize what the argument is. The clarification done by Luboš Motl that what you measure is not a property of the system under weak measurement, but a combined property of the system-meter system is more helpful, but it can be further improved. I recommend reading the article by Dmitri Sokolovski "Are the Weak Measurements Really Measurements?" Quanta 2013; 2: 50–57. DOI:10.12743/quanta.v2i1.15
There are several important points to make. First note that in the definition of the weak value you have an observable $\hat{A}$. So, when you measure weak value it is not just the issue that you measure "some property", it is good for what you are measuring  to have some plausible interpretation in connection with the observable $\hat{A}$. Sokolovski illustrates the discussion with a double slit. There the observable $\hat{A}$ has two eigenvalues $1$ or $2$, which indicate the particle passing through slit $1$ or $2$, respectively. If you take the expectation value of the observable $\hat{A}$ in the usual case of projective measurements you will find that the expectation value $\langle\hat{A}\rangle$ is within the interval defined by the minimim and the maximum possible eigenvalue. Because of that you can meaningfully interpret your measurement as saying something about the observable $\hat{A}$; in this case through which slit the particle passed.
Unfortunately, in the weak measurement case, you can get any value for $Re[\frac{\langle f|A|s\rangle}{\langle f| s\rangle}]$ ranging from $-\infty$ to $\infty$; note that the issue is not just real $Re[]$ vs. imaginary $Im[]$, so the "calibration" suggested by you cannot help you much. In the given example for pre-selected and post-selected states Sokolovski obtains a real weak value of $-100$. So, can you tell from that through which slit did the particle pass? Did it pass through slit $-100$, even though there are only two slits? Or to quote Sokolovski: 

"A broken speedometer may read $50$ mph each time the car goes at $100$
  mph, and might convince the driver, but not the traffic policemen who
  stops him for speeding. Similarly, the slit number $-100$ may come up in
  a weak measurement, but cannot be used for any other purpose, such as
  convincing a potential user that the screen he is about to buy has
  more than two holes drilled in it."

In summary, I highly recommend reading the original article written by Dimitri Sokolovski, as well as some of his previously published work on weak measurements which can be freely accessed at ArXiV:Dimitri Sokolovski.
