Physical meaning of the Lebesgue measure
There are at least two different $\sigma$-algebras that Lebesgue measure can be defined on:
- The (concrete) $\sigma$-algebra ${\mathcal L}$ of Lebesgue-measurable subsets of ${\bf R}^d$.
- The (abstract) $\sigma$-algebra ${\mathcal L}/\sim$ of Lebesgue-measurable subsets of ${\bf R}^d$, up to almost everywhere equivalence.
(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.
Your question reminds me of Hamming's quote about why he believed there is no "physical significance" to the difference between Riemann and Lebesgue integration: if there were a plane whose ability to fly depends on the distinction between Riemann and Lebesgue integration then he would not care to fly in that plane.
The distinction is mathematical, not physical. A paper that explains this is http://www.mast.queensu.ca/~andrew/notes/pdf/2007c.pdf.
This kind of distinction arises much earlier in mathematics than measure theory. For example, in the study of infinite series, we have the nice theorem that the set of real numbers where a power series converges is an interval on the real line because of inequalities on power series and the completeness properties of the real numbers. Does this tidy result have a physical meaning when the most you can ever physically measure a length (using standard length units like meters) does not even extend out to something like the 25th digit after the decimal point? I do not think so, but even if you want to talk about the value of a Bessel function at a specific number, you need to have a general conception of a power series that is converging at all real numbers -- even numbers you do not care about for physical purposes -- to have a Bessel function in the first place. The set of numbers that have at most 25 digits after the decimal point has bad algebraic and analytic properties (not a field, not complete), so the set of "physically meaningful" real numbers is not something you can base a good mathematical theory on.
To put it more simply, is there a physical meaning in the trillionth digit of $\pi$? I would say no, and I've seen statements to the effect that you don't need more than 10 or 15 digits after the decimal point of $\pi$ to measure the radius of the universe down to the width of a proton, but that does not mean $\pi$ should be considered equal to 3.141592653589793.
Really comments, not an answer to the question.
(1) Mandelbrot has some comments on "physical meaning" in his book The Fractal Geometry of Nature. He relates that before 1970 or so, whenever he would try to put a Cantor set into a paper about physics, it would be rejected as "unphysical". But nowadays, physics journals are replete with papers on fractal this and fractal that.
(It sometimes becomes humorous (and painful) when an older physicist, who never learned about Lebesgue measure, attempts to write about fractals.)
(2) Quantum mechanics. It uses Hilbert space. Even for the simplest harmonic oscillator, you use a Hilbert space such as $L^2(\mathbb R)$. It doesn't work if you restrict only to Riemann integrable functions with $\int |f|^2 < \infty$. You have to use something more general, such as Lebesgue integrable functions.
Of course I suppose you can say quantum mechanics itself has no physical meaning. But I leave that to physicists to answer. I wonder what happens if you ask a physicist whether electron orbitals are Jordan measurable...
