Is the notion of Lebesgue Measure a necessary construct for statistical physics? In chat last night a user and I were discussing the "physical" meaningfulness of the notion of lebesgue measure. In particular, we were curious as to whether physicists can "make do" without it. I mentioned that the dominated convergence theorem is needed to prove certain theorems in statistics that would be needed in areas like statistical thermodynamics, where you want to know that when dealing with a huge quantity of particles things like velocity/energy are approximately normally distributed (Central Limit Theorem). We were then surprised to find a proof the CLT that no only was free of the DCT, but formulated entirely in terms of the Riemann Integral.
My question is: Are there any specific areas in physics that rely on the notion of the lebesgue measure? (either directly or indirectly via theorems for which this notion is needed to prove). To the point of being necessary and not merely useful?
 A: Unpopular opinion here: no, you don't "need" the Lebesgue measure to do physics. You don't need any kind of functional analysis, nor any distribution theory, nor any math beyond what a high schooler knows.
None of these things are essential to describing what Nature does; the content of the postulates of quantum mechanics, or special relativity, or statistical mechanics are physical, not mathematical. It is true that you can add in fancy mathematical objects to make the postulates look nicer, as was often done decades or centuries after the fact. But that's a different thing entirely from the core task of finding out how Nature behaves.
Consider a single particle with the Hilbert space $L^2(\mathbb{R}^3)$. Absolutely nothing experimentally visible changes if I put in a momentum-space cutoff, e.g. a discrete lattice that the particle hops on, say at the Planck scale. Neither does anything change if I put the particle in a large but finite box, say, the size of the observable universe. But now we're in a finite-dimensional vector space and there's no need for fancy math.
We still need calculus, but even this can be removed; just discretize time, performing timesteps like every numerical simulation ever written. Then you're down to just elementary arithmetic. We've lost all the math, but we still have quantum mechanics, because the physical postulates of superposition, unitary evolution, the Born rule, etc. are all still intact. Similarly, in statistical mechanics, the thermodynamic limit $N \to \infty$ doesn't exist; you're always dealing with a perfectly finite number of particles, reducing the situation to classical mechanics. This setup works even in relativistic quantum field theory; it is regularly used in lattice QCD, where it produces results that can't be found any other way. 
I don't disagree that mathematical tools can be elegant and useful, but I strongly object to conflating them with the physics itself. We know how Nature works once we find the right set of laws and show they agree with experiment -- not when we write them with full mathematical rigor.
A: edit I have edited the answer to deal with some of the criticism in the comments
To the extent that the Lebesgue measure is needed to define Lebesgue integration, it is central to Quantum Mechanics: in general we require wavefunctions, as a function of position, to be Lebesgue square integrable. 
More specifically, in QM states respond to rays in a Hilbert space. Hilbert spaces are complete inner product spaces, and the Lebesgue integral is required to complete the relevant Hilbert space, see When is Lebesgue integration useful over Riemann integration in physics?
It is true that there is nothing special about the position basis, but this requirement cannot be escaped: the Lebesgue measure is necessary to define an appropriately square normalisable wavefunction in the position basis.
A: The integral was first defined by Newton as part of his method of Fluxions, ie the calculus.
Riemann put the integral on a rigorous footing using the methods of real analysis. It was quickly understood that it didn't have many good properties under taking limits. 
(To note how useful this property is one might recall one of the stories that Feynman related in his books where he would swap move the derivative operator past the integral sign).
The definition that was finally adopted was the Lesbegue integral which did - for example it had the monotone and dominated convergence theorems and the Fubini theorem on swapping limits in a double integral. Also it was capable of being defined on abstract spaces, for example groups, and in fact, locally compact groups have an avatar of the Lesbegue measure, called the Haar measure, which is the unique translation invariant measure. Finally, square integrable functions formed a complete inner product space, aka a Hilbert space.
In so far as physics is or cares to be formalised  (which isn't always the case, for example one only need note the Dirac delta function or the path integral) it has proven useful to define appropriate function spaces and so on. 
