Question (informal)
Is there an empirically verifiable scientific experiment that can empirically confirm that the Lebesgue measure has physical meaning beyond what can be obtained using just the Jordan measure? Specifically, is there a Jordan non-measurable but Lebesgue-measurable subset of Euclidean space that has physical meaning? If not, then is there a Jordan measurable set that has no physical meaning?
If you understand my question as it is, great! If not, in the subsequent sections I will set up as clear definitions as I can so that this question is not opinion-based and has a correct answer that is one of the following:
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Yes, some Jordan non-measurable subset of Euclidean space has physical meaning.
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No, there is no physically meaningful interpretation of Jordan non-measurable sets (in Euclidean space), but at least Jordan measurable sets do have physical meaning.
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No, even the collection of Jordan measurable sets is not wholly physically meaningful.
In all cases, the answer must be justified. What counts as justification for (1) would be clear from the below definitions. As for (2), it is enough if the theorems in present scientific knowledge can be proven in some formal system in which every constructible set is Jordan measurable, or at least I would like citations of respected scientists who make this claim and have not been disproved. Similarly for (3), there must be some weaker formal system which does not even permit an embedding of Jordan sets but which suffices for the theorems in present scientific knowledge!
Definitions
Now what do I mean by physical meaning? A statement about the world has physical meaning if and only if it is empirically verified, so it must be of the form:
For every object X in the collection C, X has property P.
For example:
For every particle X, its speed measured in any reference frame does not exceed the speed of light.
By empirical verification I mean that you can test the statement on a large number of instances (that cover the range of applicability well). This is slightly subjective but all scientific experiments follow it. Of course empirical verification does not imply truth, but it is not possible to empirically prove anything, which is why I'm happy with just empirical evidence, and I also require empirical verification only up to the precision of our instruments.
I then define that a mathematical structure $M$ has physical meaning if and only if $M$ has a physically meaningful interpretation, where an interpretation is defined to be an embedding (structure-preserving map) from $M$ into the world. Thus a physically meaningful interpretation would be an interpretation where all the statements that correspond to structure preservation have physical meaning (in the above sense).
Finally, I allow approximation in the embedding, so $M$ is still said to have (approximate) physical meaning if the embedding is approximately correct under some asymptotic condition.
For example:
The structure of $V = \mathbb{R}^3$ has an (approximate) physically meaningful interpretation as the points in space as measured simultaneously in some fixed reference frame centred on Earth.
One property of this vector-space is:
$\forall u,v \in V\ ( |u|+|v| \ge |u+v| )$.
Which is indeed empirically verified for $|u|,|v| \approx 1$, which essentially says that it is correct for all position vectors of everyday length (not too small and not too big). The approximation of this property can be written precisely as the following pair of sentences:
$\forall ε>0\ ( \exists δ>0\ ( \forall u,v \in V\ ( |u|-1 < δ \land |v|-1 < δ \rightarrow |u|+|v| \ge |u+v|-ε ) ) )$.
This notion allows us to classify scientific theories such as Newtonian mechanics or special relativity as approximately physically meaningful, even when they fail in the case of large velocities or large distances respectively.
Question (formal)
Does the structure of Jordan measurable subsets of $\mathbb{R}^3$ have (approximate) physical meaning? This is a 3-sorted first-order structure, with one sort for the points and one sort for the Jordan sets and one sort for $\mathbb{R}$, which function as both scalars and measure values.
If so, is there a proper extension of the Jordan measure on $\mathbb{R}^3$ that has physical meaning? More specifically, the domain for the sort of Jordan sets as defined above must be extended, and the other two sorts must be the same, and the original structure must embed into the new one, and the new one must satisfy non-negativity and finite additivity. Bonus points if the new structure is a substructure of the Lebesgue measure. Maximum points if the new structure is simply the Lebesgue measure!
If not, is there a proper substructure of the Jordan measure on $\mathbb{R}^3$ such that its theory contains all the theorems in present scientific knowledge (under suitable translation; see (*) below)? And what is an example of a Jordan set that is not an element in this structure?
Remarks
A related question is what integrals have physical meaning. I believe many applied mathematicians consider Riemann integrals to be necessary, but I'm not sure what proportion consider extensions of that to be necessary for describing physical systems. I understand that the Lebesgue measure is an elegant extension and has nice properties such as the dominated convergence theorem, but my question focuses on whether 'pathological' sets that are not Jordan measurable actually 'occur' in the physical world. Therefore I'm not looking for the most elegant theory that proves everything we want, but for a (multi-sorted) structure whose domains actually have physical existence.
The fact that we do not know the true underlying structure of the world does not prevent us from postulating embeddings from a mathematical model into it. For a concrete example, the standard model of PA has physical meaning via the ubiquitous embedding as binary strings in some physical medium like computer storage, with arithmetic operations interpreted as the physical execution of the corresponding programs. I think most logicians would accept that this claim holds (at least for natural numbers below $2^{1024}$). Fermat's little theorem, which is a theorem of PA, and its consequences for RSA, has certainly been empirically verified by the entire internet's use of HTTPS, and of course there are many other theorems of PA underlying almost every algorithm used in software!
Clearly also, this notion of embedding is not purely mathematical but has to involve natural language, because that is what we currently use to describe the real world. But as can be seen from the above example, such translation does not obscure the obvious intended meaning, which is facilitated by the use of (multi-sorted) first-order logic, which I believe is sufficiently expressive to handle most aspects of the real world (see the below note).
(*) Since the 3-sorted structure of the Jordan measure essentially contains the second-order structure of the reals and much more, I think that all the theorems of real/complex analysis that have physical meaningfulness can be suitably translated and proven in the associated theory, but if anyone thinks that there are some empirical facts about the world that cannot be suitably translated, please state them explicitly, which would then make the answer to the last subquestion a "no".
Best Answer
There are at least two different $\sigma$-algebras that Lebesgue measure can be defined on:
(There is also the Borel $\sigma$-algebra ${\mathcal B}$, but I will not discuss this third $\sigma$-algebra here, as its construction involves the first uncountable ordinal, and one has to first decide whether that ordinal is physically "permissible" in one's concept of an approximation. But if one is only interested in describing sets up to almost everywhere equivalence, one can content oneself with the $F_\delta$ and $G_\sigma$ levels of the Borel hierarchy, which can be viewed as "sets approximable by sets approximable by" physically measurable sets, if one wishes; one can then decide whether this is enough to qualify such sets as "physical".)
The $\sigma$-algebra ${\mathcal L}$ is very large - it contains all the subsets of the Cantor set, and so must have cardinality $2^{\mathfrak c}$. In particular, one cannot hope to distinguish all of these sets from each other using at most countably many measurements, so I would argue that this $\sigma$-algebra does not have a meaningful interpretation in terms of idealised physical observables (limits of certain sequences of approximate physical observations).
However, the $\sigma$-algebra ${\mathcal L}/{\sim}$ is separable, and thus not subject to this obstruction. And indeed one has the following analogy: ${\mathcal L}/{\sim}$ is to the Boolean algebra ${\mathcal E}$ of rational elementary sets (finite Boolean combinations of boxes with rational coordinates) as the reals ${\bf R}$ are to the rationals ${\bf Q}$. Indeed, just as ${\bf R}$ can be viewed as the metric completion of ${\bf Q}$ (so that a real number can be viewed as a sequence of approximations by rationals), an element of ${\mathcal L}/{\sim}$ can be viewed (locally, at least) as the metric completion of ${\mathcal E}$ (with metric $d(E,F)$ between two rational elementary sets $E,F$ defined as the elementary measure (or Jordan measure, if one wishes) of the symmetric difference of $E$ and $F$). The Lebesgue measure of a set in ${\mathcal L}/{\sim}$ is then the limit of the elementary measures of the approximating elementary sets. If one grants rational elementary sets and their elementary measures as having a physical interpretation, then one can view an element of ${\mathcal L}/{\sim}$ and its Lebesgue measure as having an idealised physical interpretation as being approximable by rational elementary sets and their elementary measures, in much the same way that one can view a real number as having idealised physical significance.
Many of the applications of Lebesgue measure actually implicitly use ${\mathcal L}/\sim$ rather than ${\mathcal L}$; for instance, to make $L^2({\bf R}^d)$ a Hilbert space one needs to identify functions that agree almost everywhere, and so one is implicitly really using the $\sigma$-algebra ${\mathcal L}/{\sim}$ rather than ${\mathcal L}$. So I would argue that Lebesgue measure as it is actually used in practice has an idealised physical interpretation, although the full Lebesgue measure on ${\mathcal L}$ rather than ${\mathcal L}/{\sim}$ does not. Not coincidentally, it is in the full $\sigma$-algebra ${\mathcal L}$ that the truth value of various set theoretic axioms of little physical significance (e.g. the continuum hypothesis, or the axiom of choice) become relevant.