Mean Gravitational 'Potential' Energy in Space Imagine an arbitrary point in space. It is within the gravitational 'potential' of every mass (although billions of ly away) in the entire universe. 
Since every mass adds a tiny fraction, what is the total gravitational 'potential' energy in this point?
Edit:
Let point masses be located distance $r_i$ from the point and have masses $m_i$, then the 'potential' is
$$\Phi = - G \sum_{i} \frac{m_i}{r_i}.$$
I'm looking for this value averaged over all points in space. How does this depend on the shape of our universe or can we measure it?


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*For example gravity on my location is given by
$$ 9.81 m/s^2 \text{(earth )} + 6 mm/s^2 \text{(sun)} + 200 pm/s^2 \text{(milky way)} +  ?  \text{(rest of the universe)}.$$
 A: The universe cannot be described at cosmological scales using newtonian physics. We have to use general relativity.

How does this depend on the shape of our universe or can we measure it?

Ideas like the "shape of the universe" don't make sense in newtonian gravity, so for this reason as well we need general relativity in order to answer your question.
In general relativity, a gravitational potential exists only for static spacetimes. Realistic cosmological models are not static, and therefore we can't define a potential for them.
A: 
Since every mass adds a tiny fraction, what is the total gravitational
  potential energy in this point?

There is no unique value of gravitational potential energy, or gravitational potential, at a point in space. It is always measured relative to another point in space.
A mass $m$ in space has gravitational potential energy $U$ with respect to another mass $M$ where the two masses are separated by a distance $r$ of
$$U=\frac{-GMm}{r}$$
Since gravitational potential energy is inversely proportional to $r$ it make sense to choose the zero of gravitational potential energy of $m$ with respect to $M$ for infinitely large $r$. 
However, to the extent that there may be a net gravitational force acting on $m$ due to all the masses $M$ acting on $m$, the mass $m$ will experience an acceleration. If we were able to measure the change in velocity of $m$ between two points, we could determine the difference in gravitational potential energy between the two points since the loss in gravitational potential energy between the points will equal the increase in kinetic energy between the points, neglecting any friction in space. 
Hope this helps.
A: It seems to me you are asking about gravitational potential, not gravitational potential energy, since gravitational potential energy is property of some object and not space.
First of all, potential must be specified against some reference point. The Universe as we understand it today is homogeneous and isotropic at large scales. Therefore at large scales all points of space are equivalent and no matter which reference point you will choose (if you average the inhomogeneities at small scales) the difference will be zero. So the distant sources do not matter for the gravitational potential, all that matters is only close surrounding.
You can imagine it this way: the potential tells you how much work you need to do to move some object of unit mass from the reference point to the point in question. But since universe is everywhere (on the large scales) the same and every point on the path is as good as previous one, no work is needed. 
