New paper "Non-Smooth Integrability Theory" has been released

Um... I asked for the typos to be fixed properly, but they haven't been...
Just to mention, in Lemma 1, 'every' has become some mysterious symbol 'ever"' in one place, and later, inside the proof on page 24, where it should be $${t_2^+(y',z')}$$, it says $${t_2^+(y',c^*z)}$$.
Other than that... I think it's correct... I think...

Oh, and I'll touch on the content as well; it's about integrability. As for the problem, given a candidate demand function $${f(p,m)}$$, we address when it can be considered a demand function and how to construct a utility function in that case (this is Proposition 1). However, if I just say that, it feels no different from a previous paper by Hosoya (2017, J. Math. Econ), so to explain what has changed: I do not assume that f is differentiable. Instead, I assume it is locally Lipschitz.
To elaborate on this point, I was thinking of applying integrability to the problem of statistical estimation of utility functions. In other words, while there is no data corresponding to u, there is purchase data corresponding to f, so if u can be calculated from f, then an estimate of u can be obtained from an estimate of f. With this in mind, I was organizing the existing assumptions, but there were various problems... and that's how it is.
First, if f is continuously differentiable, the Slutsky matrix can be defined, and various analyses can be performed using it. However, the problem of estimation incorporating conditions that guarantee the continuous differentiability of f seems to be very difficult, and there are almost no econometrics papers dealing with this. Therefore, this assumption cannot be called practical.
On the other hand, if f is only continuous, there may be two continuous utility functions u and v that represent different orderings such that $${f=f^u=f^v}$$. This means that u cannot be calculated from f, so it cannot be used for the estimation problem of u. This problem disappears when f is locally Lipschitz with respect to m. I wish I could have discussed it under this assumption, but it was just too difficult... I couldn't prove the part corresponding to step 4 of the proof of Theorem 1 in the above paper under that assumption. So, the current paper is a compromise where I simply assume it is locally Lipschitz from the start.
By assuming it is locally Lipschitz, we can use a theorem called Rademacher's theorem, and f becomes totally differentiable at almost every point. Therefore, the Slutsky matrix can also be defined at almost every point. The main point of the above paper is to try to do something using this. In conclusion, first, the fact that f is a demand function is equivalent to the symmetry + negative semi-definiteness of the Slutsky matrix 'at almost every point'. Furthermore, the following partial differential equation corresponding to Shephard's Lemma

$$
\nabla E(q)=f(q,E(q)),\ E(p)=m
$$

also necessarily having a globally concave solution is equivalent. The latter is important, and by using this, we can prove that if the limit of a sequence of locally Lipschitz demand functions in the sense of broad uniform convergence is also locally Lipschitz, then it is also a demand function. In other words, the set of demand functions has a certain kind of closedness. Taking it a step further, if we construct a space of demand functions that are uniformly Lipschitz on any compact set, this set becomes compact.
...Compact? You might think, but it is compact. At first, I thought I could only show it was complete, but... it was compact. After all, having a uniform Lipschitz constant means they are equicontinuous... so it was easily shown using the Arzelà-Ascoli theorem.
And, if we add a few more assumptions, we can say that for this sequence of demand functions, the sequence of corresponding utility functions converges to the utility function corresponding to the limit demand function. Thus, we found that if the estimate of the demand function is sufficiently close to the true value in the relatively decent topology of uniform convergence, the estimate of the corresponding utility function is also sufficiently close to the true value. Happily ever after.

...Are you satisfied with just this much?

Since compactness has emerged, here is an additional result. Let's consider a sequence of demand functions that converges pointwise. From compactness, if we take a subsequence, it converges uniformly. By using this, as long as the sequence of demand functions converges pointwise, the sequence of corresponding utility functions converges in the sense of broad uniform convergence —a tremendous result has emerged. I'm surprised.
Besides that, well, it contains a large number of examples and counterexamples, so if you are interested, please take a look. It is definitely, without a doubt, the work I am most confident in among my micro papers in the last few years!
That's all for the advertisement.