What's the meaning of distant measuring? this is the context:

Einstein, Podolsky, and Rosen concluded that there are two possibilities in the case of the spinning particles. Either (a) there is some mysterious, nonlocal interaction that is different from any other branch of physics that explains how Bob’s particle is affected by Ann’s measuring her particle. If this was true, our naive notion of space where distant objects and measurements are independent of each other is wrong.

I don't get this part:

If this was true, our naive notion of space where distant objects and measurements are independent of each other is wrong.

Can anyone explain this part from the physics point of view? what is the relationship between these two? 
thanks in advance.
 A: The EPR paradox is actually not a "paradox", in the sense that it is understood within quantum mechanics. Both particles are described by a single wavefunction, in which both particles are entangled. When you make a measurement at either end, you cause the collapse of the whole wavefunction, which is why also the other end can see the result of said measurement.
The "paradox" comes up when you try to "make sense of this" classically. Classically, that is, you would expect that the two particles infinite separation now behave as independent objects. Every interaction$^\dagger$ in nature has some finite range, think of electric fields from a point charge decaying as $\propto1/r^2$ etc. So at infinite separation, you would (classically) expect that the particles cannot affect each other's dyamics or what not.
The fact that you see there is a residual "interaction" between the two particles, has to effects:


*

*Our notion of space is wrong. You'd expect that if separated by a distance $> ct$, $c$ being the speed of light, the first particle cannot instantaneouosly influence the other one. But if an instantaneous cause-effect phenomenon is observed, then this notion of "space" being the the only separation between the particles, controlling how much time light travels and/or the decay of interactions, is wrong. We must accept non-local phenomena (i.e. causally related events happening instanteously at different positions). This is what Einstein referred to as  “spooky action at a distance.” 

*The particles possessed the measured quantity all along. I.e. they gained it while they were in causal contact at time $t=0$, and they just retained it while travelling. When you mesaure one, you are merely opening a box and unvelining a note with information that was written in the (causally connected) past. This is OK in classical physics, where a particle has a specified momentum/spin (etc) at all times, but it is not ok with (the Copenhagen interpreation of) quantum mechanics. Attempts to make the wavefunction $\psi$ have pre-determined values that do not obey the statistically distribution upon measurement are called hidden-variable theories and have been shown to be inconsistent - see for instance Bell's inequalities.
Both points are wrong. So far causality and locality have agreed with all experimental observation and are hence taken to the "correct". Quantum mechanics is also correct, that is it has survived all experimental verification.
The solution to the "paradox" is that we cannot think of this situation in terms of classical physics. The two particles will never become independent, because of quantum entanglement.

$\dagger$: from localised charges (i.e. not infinite sheets or infinite lines of charge)
A: The "notion of space where distant objects and measurements are independent of each other" is a property know as locality, which is roughly the idea the strength or significance of physical effects should fall off with distance and time. In the case of an EPR experiment, violation of locality appears if we imagine that there is a long-distance interaction that causes the state of one particle to be determined with certainty as soon as the state of the other particle is measured.
As SuperCiocia points out, one way to kind of get around this is to just insist that the two particles are not actually two distinct systems, but instead a single system that is distributed in space. In my opinion this is indeed a better way to think about the situation, but we shouldn't be blind to the fact that it is still quite surprising: we no longer talk about it as a long-range instantaneous interaction, but having a single object whose properties may be distributed over an arbitrarily large (or in the case of EPR, long) region of space without attenuation is a dramatic violation of our classical intuitions (not saying that's a problem, but it's worth observing!)
It is also always important to note, when we talk about EPR experiments, that they cannot be used to send information faster than the speed of light, and in no (known) situation do they violate special relativity (see Does entanglement not immediately contradict the theory of special relativity?, and links therein.)
