A banal lemma from measurable functions

[wrobel]

For a lecture, I needed to state and prove a lemma. See the text. I wrote the proof and just ran it through an AI for good measure. And then all hell broke loose! :) Did I actually write an incorrect proof?

 

[renormalize]

Did I actually write an incorrect proof?

I don't know the answer, but I display below your short PDF for easier reading:

 

[pbuk]

It is not clear where your proof ends and "all hell breaking loose" starts, but it does seem to me that the words "not equal to a constant almost everywhere" in the lemma can be interpreted in two ways: "[not equal to a constant] almost everywhere" (which is what I think you meant) and "not [equal to a constant almost everywhere]" (which is what has been proved).

Perhaps you should rewrite the lemma as "almost everywhere not equal to a constant".

 

[wrobel]

words "not equal to a constant almost everywhere" in the lemma can be interpreted in two ways

In the literature, I have only seen one interpretation of these words: for any constant c the statement 'f(x)=c a.e. ' is wrong. See for example

Eigenvalue theorems in topological transformation groups

 

[pbuk]

In the literature, I have only seen one interpretation of these words: for any constant c the statement 'f(x)=c a.e. ' is wrong.

Maybe so, however you seem surprised by something (I am still not sure what) an AI has done. Is it possible that the AI is not as familiar with this context as you are?

 

[wrobel]

Gemini said that my proof was full of errors. In the course of our further conversation, I convinced it that there were no errors. But you can convince an AI of anything, which is why I created this thread. Unfortunately, I deleted that chat.

 

[Matterwave]

The shape of the proof seems ok to me. There are points of the proof which I myself can't vouch for the rigor (the step $f_N=f_{N+1}=...$ doesn't seem to follow for me but I could be missing something) and I think there's a few quirky notation ($E_n=\{|f(x)|<n\}$ I think you mean $E_n=\{x\,|\,|f(x)|<n\}$?) but the high level logic seems ok and I can't think of a counter example. Of course, I'm not even a mathematician so maybe my opinion here is not valuable lol.

When you talk to AI about it, do you use words or images? It's probably best to use words as that's where most of the training goes. Also, as you note, it is indeed quite easy to convince AI to just agree with you. I would also only try this exercise with frontier models (Opus class). Last, it's best to make statements more concrete before giving to AI. Esp the ambiguity noted by pbuk. That was also ambiguous to me when I first read the lemma.