Quantum mechanics and points in space

There are two different pictures of space: space is made of points or space is made of regions. I will argue that quantum mechanics naturally suggests the latter view.

Consider a single-particle system. Its quantum state can be representated by a wavefunction Ψ, which is a square-integrable complex-value function. This seems to go nicely along with the idea that there is such a thing as space, and each point in space has the property of having such-and-such a value of the wavefunction Ψ. (And in a multi-particle system, tuples of points are related to a value.) But that misses a subtlety. While a square-integrable function Ψ represents a quantum state, any two square-integrable functions Ψ₁ and Ψ₂ that agree outside of a set of measure zero are taken to represent the same quantum state. The measure of a (set consisting of a) single point is going to be zero. So there is no a physical fact of the matter as to the value of the wavefunction is a particular point.

This seems to me to cohere a little better with a view on which space is built up out of extended regions rather than points. While there will be no fact of the matter as to what "the" value of the wavefunction is at a point x, for any extended (at least in the sense of being nonzero-measure) but bounded region R of space, say a ball or cube, there will be a fact about what the average value of the wavefunction over R is (functions that are square-integrable are locally integrable). Moreover, one can recover the value of the quantum state from the values of such averages over, say, all balls. (If space is potentially subdivisible but not actually subdivided, some of these balls will be potential regions of space, and the average of the wavefunction over such a ball may be a dispositional property--the average it would have if space were divided so as to have that ball as a region.)

This is not a knock-down argument against views on which space is made of points. One could say that space is made of points but deny that quantum states have values at single points. Or one could say that wavefunctions that differ on a set of zero measure represent different in principle empirically indistinguishable quantum states. The latter is, I think, unattractive. We should avoid positing an infinite number of empirically unobservable degrees of freedom in physics.