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I am studying applied math and I currently got stuck on proving that a function, which emerges in a model is measurable (Borel functon), so we can integrate it. I know, that there are examples of non-measurable sets w.r.t. the Lebesgue measure (Vitali set) so its characteristic(indicator) function will be non-measurable too. But my question is the following:

Is there an example of real-life application where the verification of measurability really matters? For example a mechanism, for which we can try to derive its behavior (for example, stability properties), ignoring measurability check (just writing integrals mindlessly), but which do not follow our prediction exactly because we assumed some function within the model to be measurable but it is actually not?

In the other words, I want historical "proof" of importance of this particular type of mathematical correctness.

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    $\begingroup$ As you need the axiom of choice to have non-measurable things for the Lebesgue measure, I doubt mesurability of classical functions for the Lebesgue mesure has real life implication. It's different for stochastic process where measurability really matter $\endgroup$ – Tryss Jun 19 '15 at 0:49
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Caveat: I give this more as a general defense of why we must check that our functions have the properties we like, when dealing with a physical model. The reason being that I don't know a good example where measurability proper could cause issues. Nevertheless I believe the idea will be quite the same.


I think it's more of a measure (wink) that has to be observed in order further examine your model with all correctness. Perhaps you postulate (as doing otherwise would lead to few conclusions, if any) that the "primordial" functions (position, density, etc.) are measurable, and more things if you like. However you might, by the principles that constitute your model, arrive at something such as, say, the Laplacian of a function $$\Delta v = \frac{\partial ^2 v}{\partial x_1^2} +\cdots + \frac{\partial ^2 v}{\partial x_n^2}$$ Will this be measurable where the partials exist? In this case the answer (I'm almost positive) is yes. But, for example $\Delta v$ will not necessarily be differentiable any further. Or even harder: if our basic principles give that $\Delta v$ must satisfy some relationship, does this mathematically imply that $v$ is as we want it to be? There's a million bucks for the answer (In this case "what we want it to be" is more than measurable). Indeed, differentiation generally provokes a terrible loss in regularity, which is bad news for the study of (models based on) partial differential equations for one (and not so much for ordinary DEs).

But forget differentiating, is the sum of measurable functions measurable? What about products, maximums, limits, etc. of measurable functions? We manipulate and twist and bend and throw about functions that may have originally been "good" (measurable, square-integrable, continuous, etc.), but we must make sure that the end result isn't too disfigured. In most models that I know of, a routine verification is whether certain some form of regularity holds, be it continuity, smoothness, or something related. As for measurability, this will probably arise when dealing with stochastic systems, but I don't know a lot so I can't say more.

In summary: we often model physical observables "indirectly" by prescribing a mathematical relationship between them (usually, an equation). We therefore have to check that, conversely, such a relationship will always turn out sufficiently regular functions. Otherwise it might precisely be the case that a certain observable is not, for instance, measurable, which we obviously don't want.

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