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The problem with Riemann integration of functions like ${\chi }_{ℚ}(x)$ (a function which takes the value of 1 if $x\in ℚ$, and 0 otherwise) is usually used to show the need for defining the Lebesgue integral. Since I don't have much practical experience with actual science and research, I have a question: are measure zero sets physically relevant (meaning that they are found as objects of interest among physicists or play an important part in some physical models), or are they used just to provide "pathological" examples and describe the need for some improvements in mathematics, like the Lebesgue integral?

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closed as too broad by ACuriousMind, John Rennie, Qmechanic Sep 1 '15 at 14:14

Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. Avoid asking multiple distinct questions at once. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

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    $\begingroup$ This question seems like a list question. $\endgroup$ – ACuriousMind Aug 31 '15 at 23:34
  • $\begingroup$ Not necessarily, I'm looking for just one good example. Of course, the more the better, but it's not obligatory. $\endgroup$ – Soba noodles Aug 31 '15 at 23:38
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    $\begingroup$ Yeah, but so there's not one correct answer, there are potentially dozens of them. This is the definition of too broad. Not to leave you hanging though, look at $L^2$-spaces in quantum mechanics. $\endgroup$ – ACuriousMind Aug 31 '15 at 23:39
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are measure zero sets physically relevant (meaning that they are found as objects of interest among physicists or play an important part in some physical models),

They come up naturally. For instance if you study the actual dynamics of putting an eigenstate of $\sigma_x$ into a z oriented Stern-Gerlach device you'll see the Schrödinger equation predicts the upper half of the beam goes up and the lower half goes down (and that the spin state changes so that each branch has a spin that is an eigenstate of $\sigma_z$).

But what about the past of the beam exactly on the middle? It is a set of measure zero (its a surface in 3d) and so they happen with probability zero (which means they can happen but the expected frequency for lots of observations is less than any positive number).

And they come up in the regular pourtreal of square integrable functions (they they are only functions up to sets of measure zero).

But the latter example is a bit far fetched. For instance we could just start with, say, the energy eigen functions of the Harmonic Oscillator as a fine orthonormal set and then we can take their span and then take the closure and call that our Hilbert Space. Or even be explicit about how we take the closure and define the Hilbert Space to be the set of equivalence classes of Cauchy sequences of things in the span of the energy eigenfunctions. And not that here we never ever need to bring up Lebesgue integration. The energy eigenfunctions are perfectly fine Riemann integrable functions they have a perfectly fine inner product and so we can have Cauchy sequences of them and then take two sequences to be in the same class is their difference is a sequence that goes to zero in the norm from the inner product.

We never had to do a Lebesgue interval, never needed measure theory, and didn't have worry about measurable sets.

or are they used just to provide "pathological" examples and describe the need for some improvements in mathematics, like the Lebesgue integral?

As you can see above the Lebesgue integral isn't an improvement. Historically the problem was that mathematicians simply made up some axioms that made it appear there were weird sets (think Banach-Tarksi decomposition) so they either had to give up their axioms and use other axioms or else learn to avoid those sets. So they made measure theory to avoid those sets. And then Lebesgue integration to use it. To make more functions integrable. But we didn't ever need to have so many functions in the first place. You can get the Hilbert Space with out worrying about the vectors in the Hilbert space being an equivalence class of square integrable measurable functions. They can be equivalence classes of Cauchy sequences of useful functions.

As a Hilbert Space they are equally good. You don't have to worry about weird functions. You can compute your predictions to any Kevel of accuracy without them and even have the same mathematical object (the Hilbert Space) without them.

Mathematicians say something is good if they can prove theorems with/about it using their favorite axioms and favorite deductive methods and sometimes if it uses their favorite theorems.

Physicists say something is good if it allows us to predict and understand the universe.

These two goals are not always the same. In fact, often they are not. Physicists might be more aware of the times when they match because of selection bias.

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  • $\begingroup$ So, for most practical purposes (analysis of most experimental data) I don't have to worry about measure zero sets because in most cases I'm only interested in the normal convergence of my results to the predictions instead of uniform or pointwise convergence? After all, measurement results come with an error margin, so speaking of isolated points in the continuous distribution with a different / discontinued value doesn't make much sense. $\endgroup$ – Soba noodles Sep 1 '15 at 7:33
  • $\begingroup$ Anyway, your answer is the most helpful and insightful one so I'll mark it as accepted. $\endgroup$ – Soba noodles Sep 1 '15 at 8:08
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In quantum field theory, one often considers meager subsets of the naively defined space of possible states. Indeed this is one of the Wightman Axioms: the Hilbert space of physical states is separable (has countable basis).

This is a manifestation of the intuitive assertion "we must only put a finite number of particles into a finite quantisation volume".

Just take a Fermionic Fock space for a finite quantisation volume, so that there are countably infinitely many plane wave as the "modes". Then an arbitrary basis member is of the form:

$$\left|\left.0,1,1,0,1,1,1,0,1,0,\cdots\right>\right.$$

that is, a countably infinitely long string of 0s and 1s showing which modes are filled and which are ground state. This set is mapped bijectively to the interval $[0,\,1])$ - its members are simply binarys expansions of numbers in $[0,\,1)$. So it's uncountable.

The "physical" states are those with a finite number of particles, so they are the dense subset $\mathbb{Q}_2\cap[0,\,1)$: the set of rationals in the interval with finite binary expansions. These are clearly a meager (zero measure) set.

You may also be interested in the Physics SE Question Separability axiom really necessary?

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