1. The measurement of the points "A" and "B" themselves, forming the extremities of the path.
2. The measurement of the path length squared, |X|², between A and B.  Keep in mind this measurement is important to how the space will be defined.  We can not assume a straight line connecting A and B or in other words a linear path can not be assumed in space between two nearest points.  (This point is beautifully elucidated by the General Theory of Relativity.)

    The area under the each curve is unity.  We can also think of a scenario where there is only one observer and three different paths are being measured.  Path-A requires only a single measurement and essentially it is a zero-entropy process (not achievable).  Path-B is a finite entropy process, hence points A and B can be measured precisely (with observer's maximum capacity) but |X|² varies.  And Path-C is an infinite entropy process where neither points A and B nor the connections between them can be measured precisely (observer in an inertial frame, or classical Newtonian mechanics).

    The path AB can be represented by the closed interval [0,1] for Obs_C or Obs_M.  However per Obs_i criterion the interval is open and represented by ]0,1[.
    
    To summarize, we have defined the problem of measurement.  The definition is independent of the observer's measurement-metric.  We have discussed the existence of the curvature in j-space which must be followed by (t, x, y, z, m, q) or the space-time metric corresponding to the macroscopic observer in Λ_∞ plane.  We have also discussed the statistical nature of the measurements made by the observers of various capacities. We note that the actual shortest path between two points was never determined and it was not part of the measurements made by the either observer.  These are important issues needed to be resolved before discussing the EPR paradox and the "Copenhagen Interpretation" of Quantum Mechanics.

Why are we measuring the quantity |X|² ?  Why not measure the path-length |X| itself ?

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1. The Gaussian or Normal distribution along with Cauchy and Lèvy distributions, is known as a stable distribution. The stable distributions follow the linearity property. For example, two Gaussian distributions when derived independently for the same random variable and then combined, will provide another Gaussian distribution. These distributions are not analytically expressible i.e. the relationship between the points can not be expressed in form of an equation.  In j-space such distributions represent PE1_j measurements or stable structures, in Λ_∞ plane. There are distributions such as Chi-Square (bounded at zero), who approach Gaussian distribution under asymptotic limits and hence stability.

    We realize that we are associating linearity and hence "addition" to the "stability of the structure" in j-space. This is the first time we have introduced an arithmetic operation in our measurement space and it certainly did not come from the analytical expressions. We can also introduce "subtraction" based on linearity for the low-information or high quantum number states. However we still do not know about operations such "multiplication" and "division". The VT-Symmetry is a requirement for the arithmetic operations in j-space.

2. The measurement of Obs_C  (v²/c² ≈ 1), are described by Pauli's spin matrices. We need Hamilton's anharmonic coordinates to understand their effect (i.e. finiteness), on the physical measurements made by the macroscopic observer Obs_M in Λ_∞-plane.