May I include results from a collaboration in my PhD thesis?
I had a similar situation in my thesis, in math. The third chapter was based on a work that was jointly done with another researcher, a professor from another university. (In the end the paper we published was even with two other researchers, as we decided to join our efforts.)
I had to ask the researcher for permission to include our joint results. This is a no-brainer (though of course he was happy to let me do it). When I sent out the thesis to my committee (of which he was part), I also sent him a copy beforehand and asked him if he was okay with what I had written, as they were his results too and he might want to check that I didn't mangle them before releasing them into the wild. My advisor also was on board from the start, but I imagine that you have to ask for permission if this isn't the case.
I had to include acknowledgments for this in my thesis. Basically, I wrote a paragraph at the beginning of the chapter, stating "this chapter is based on joint work with [...]". Again, this is a no-brainer. I cannot claim all the results are due to me if they aren't. However, I didn't say anything of the sort "I did X% of the work and the other researcher did (100-X)% of the work", this isn't how it works, and it would have been in rather poor taste. At least in mathematics, it's expected that co-authors contribute roughly the same amount of work to a joint paper, but it's not an exact "50-50" (or 33-33-33, or 25-25-25-25...) distribution.
The whole thesis had to be written by me, in my own words, including the results and the proofs that were actually due to the other professor. Moreover, I obviously had to perfectly understand, and be able to explain, and defend, all the results, as if they were mine.
All of this was explained to me by my advisor. It's possible that regulations in your university say something different. The essential part, I believe, which is only hinted at in the steps above, is to ask your advisor what to do exactly.
It is completely normal to include in your thesis work that was done in collaboration. Almost every PhD thesis in STEM subjects includes at least some collaborative work, simply because your PhD itself is a collaboration between you and your advisor.
Usually, you just need to clearly state what work in your thesis was done in collaboration and with whom. It may also be expected that joint work should be put in your own words, rather than just copied from the paper. That's good practice anyway, since it's a great way to make sure you really understand the work. And you'll need to understand it because you can be examined on anything that's in your thesis.
Check your individual university's regulations for the precise details. And ask your advisor – this stuff is literally their job.
Assume that while being a student, you collaborate with Prof. X, and you write a paper together. In this paper, the work was split 50-50, so that half of the results belong to Prof. X.
That simply isn't how collaboration works in mathematics. The results "belong to" all the authors, because they came from a creative process that heavily involved all of them. Of course, some parts of the work will be solely by one person sitting in their office with a cup of coffee until they figure it out on their own, but it's mostly not like that. And usually, even these "solo results" receive at least some contributions from the other authors as the paper gets written and rewritten. And, very often, even if it was Author A who came up with the actual proof, Authors B and C were part of the process that led to that particular technique being chosen, and so on.
Of course, there are some exceptions. Sometimes, an author joins a collaboration late because their expertise on some area is needed. In computer science, papers are often released first as a shortened conference version (which is a peer-reviewed publication, unlike in mathematics) and later, the full version appears in a journal. I'm aware of papers where extra authors have been added to the journal version because of specific contributions they made after the conference. I guess the closest analogue in mathematics would be a paper going up on the ArXiv and then acquiring extra authors. If an author in that situation was a PhD student, I'd expect their thesis to include only a high-level summary of the material that existed before they joined the project.
The fundamental rule is honesty. If you can put your hand on your heart and say that you made significant contributions to the work and you're clear about who else contributed, it's almost certainly fine. If you're including big chunks of stuff that you made no significant contribution to, I'd hope you already know that it's wrong to say "I did this."
The short answer: Yes. However, it may depend on several details. In my case, I did include results from a collaborator in my thesis and that was generally accepted. I was however asked to provide an acknowledgement section where I had to clarify which results were done by me and which by the collaborator. Also, my thesis was in organic chemistry so things might be different in mathematics. So while I can't say if it is common or not in mathematics, I can tell you that it was allowed in my case.