Measuring potential energy and potential energy differences In reference a discussion on gravitational potential energy $U$ (available a https://physics.stackexchange.com/questions/287292/is-energy-relative-or-absolute-does-gravity-break-the-law-of-energy-conservation), we can set $U(R)$ equal to any value. We can set $U(R)=0$ or choose that $U(r=\infty)=0$.
Do we just arbitrarily set that up? Also why "can" we only measure potential energy differences exactly? That is not true for kinetic energy, I would say. What is the issue with measuring $U$ only.
Energy, in general, is relative and only energy differences are not relative irrespective of the reference frame.
Thanks.
 A: For the discussion, I will talk only about non-relativistic cases and conservative systems.

Do we just arbitrarily set that up?

Yes! Of course.

Also why "can" we only measure potential energy differences exactly?

The line integral that defines work along curve $C$
takes a special form if the force $\mathbf{F}$ is related to a scalar field $Φ(x)$ so that
$$\mathbf{F}=\nabla\Phi$$
In this case, work along the curve is given by
$$W=\int_C\mathbf{F}\cdot d\mathbf{x}=\int_C \nabla\Phi \cdot d\mathbf{x}$$
which can be evaluated using the gradient theorem to obtain
$$W=\Phi(\mathbf{x}_B)-\Phi(\mathbf{x}_A)$$
Traditionally potential energy is chosen to be $U=\Phi$.
That's the mathematical proof for why there is a difference involved.
You can prove this with contradiction, Suppose there exist a function $\Phi(x)$ which is an absolute value of potential energy at a point.
Then this says that the work done by the particle to travel from any point $x$ to some point $x=a$ is the same. Now, this also means that work done by the particle to travel from $a+\epsilon$ to $a$ is the same as from $\infty$ to $a$. That's certainly not true! (Think why?).

That is not true for kinetic energy, I would say. What is the issue with measuring U only?

Kinetic energy depends on the magnitude of velocity so that
$$K=\frac{1}{2}m(\mathbf{v}\cdot \mathbf{v})$$
Because the distance covered while applying a force to an object depends on the inertial frame of reference, so depends on the work done. Due to Newton's law of reciprocal actions, there is a reaction force; it does work depending on the inertial frame of reference in an opposite way. The total work done is independent of the inertial frame of reference.
Correspondingly the kinetic energy of an object, and even the change in this energy due to a change in velocity, depends on the inertial frame of reference. The total kinetic energy of an isolated system also depends on the inertial frame of reference: it is the sum of the total kinetic energy in a center of momentum frame and the kinetic energy the total mass would have if it were concentrated in the center of mass. Due to the conservation of momentum the latter does not change with time, so changes with time of the total kinetic energy do not depend on the inertial frame of reference.
By contrast, while the momentum of an object also depends on the inertial frame of reference, its change due to a change in velocity does not.
