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There is a set called Vitali Set which is not Lebesgue measurable.

Analogously, there also exists a Vitali set $Y$ in $\mathbb R^3$ which is a subset of $[0,1]^3$ and $|Y\cap q|=1$ for all $q\in \mathbb R^3/\mathbb Q^3$. However, I'm curious about if it fulfilled a kind of isotropic uniform medium, let this isotropic uniform medium has density $\rho$, and put it on a electronic scale to weigh, what reading can we get? Note that $m_Y=\rho V_Y$ but $V_Y$ seems to be undefined... So it seems we cannot get any real reading. But on the other hand, since we are using a electronic scale, it also seems we must get a reading...A paradox?

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The existence of Vitali sets relies on the axiom of choice, which makes the question a duplicate of this Phys.SE post. Also related: physics.stackexchange.com/q/20370/2451 –  Qmechanic Dec 12 '12 at 16:20
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There's no paradox. We are on physics stackexchange, not mathematics stack exchange. Non-measurable sets are purely mathematical concepts that cannot be physically instantiated. Any medium in our universe is either made out of particles that are discrete or fields which, as far as we know, can be modeled as being continuous in our 4 dimensional space-time. There is no way to construct a non-measurable set in this universe.

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If an unmeasurable object existed, which law would it go against? –  Popopo Dec 12 '12 at 17:41
It could not be made of matter since matter is made of atoms and particles and it could not be a field since fields are continuous. So if you discovered such an object it would be a new form of substance which would not fit into any law of physics that we currently understand. So in a sense such an object would violate all of our laws. –  FrankH Dec 12 '12 at 19:43
Well, in newtonian mechanics objects are systems of mass points. So my question is about a Vitali set fulfilled with mass points. It sounds possible to exist. –  Popopo Dec 13 '12 at 3:42
But it is a FINITE number of points, immeasurable sets have an uncountably INFINITE number of points. So they are not comparable. –  FrankH Dec 13 '12 at 4:14
What about a mass point system contains countable infinite many mass points? It sounds strange that a mass point system can contains either finite or $\beth_1$ mass points but not to $\aleph_0$ which is actually between them. –  Popopo Dec 13 '12 at 5:46
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