Collapse isn't part of quantum theory since it isn't compatible with quantum mechanical equations of motion such as the Schrodinger equation. If collapse happens then it will require a modification of quantum theory such as the GRW theory
https://arxiv.org/abs/0710.0885
What happens instead is that quantum systems evolve by local equations of motion so the state (and the probability current) changes continuously. A measurement is just a process that copies information from one system $S$ to a measurement device $M$. In general there are multiple versions of $S$ one for each possible measurement result and there is a corresponding version of $M$ for each version of $S$. The multiple versions of $M$ undergo a process decoherence that prevents interference between then and each version evolves approximately according to classical equations of motion:
https://arxiv.org/abs/quant-ph/0306072
So $M$ has a particular state relative to each version of $S$, which is called the relative state:
https://arxiv.org/abs/2008.02328
This way of looking at quantum theory is commonly called the many world interpretation, but since collapse would require a modification of quantum theory it should just be called quantum theory:
https://arxiv.org/abs/2205.00568