I am trying to read Hugh Everetts thesis about the universal wave function, which is now called the many worlds interpretation of QM.

At page 53 ff. he discusses measurements. He talks about a system composed by two state functions, saying that the two subsystems can be considered as measuring each other, using two operators A(t) and B(t) representing those measurements. The he writes (p. 54):

I don't understand this at all. To me, a measurement is instantaneous. You apply a measuring apparatus, read it, and you are done. The short time this procedure takes is negligible, says my intuition. Yet, Everett is talking about a process where the time tends to ∞.

Haven't read the paper but I think he's allowing for decoherence time. The "short time this procedure takes" is not negligible, for his purpose. When you make a measurement you put the two systems in correlation. For instance, the spin states of a particle become correlated with the dial positions on a SG apparatus. That takes some time, of course: the decoherence time. In practice it's very quick, but once the system has "settled down" it will remain like that forever (if undisturbed). So instead of getting into the details of defining a "sufficiently long time" to make off-diagonal elements "close to zero", i.e. negligible, he just lets it go to infinity.

You mean that MWI ("this view") doesn't correspond with your intuition. Nor mine, on the face of it. Evidently there's one world only, and alternatives that might have happened, don't. But consider the wavefunction, which consists of superposed possible outcomes. Only one of them occurs: what happens to the others? In normal QM they simply disappear - "collapse". MWI says they continue to evolve according to Schrodinger's equation (or similar formulation). From a certain point of view, it can be argued that's more intuitive than "collapse".

I think Everett had a good idea, a great example of "thinking outside the box". Unfortunately the devil's in the details. For one thing, there's no good alternative to the Born rule in MWI, apparently. In spite of that, we can still admire Everett's creative insight.

I find it a little confusing, as well, but what I think he might be getting at is that a measurement involves an interaction that leaves a persistent, permanent record. If an electron passes by me, then I am temporarily affected by the electron in a way that may depend on the electron's position, or spin, or momentum. But if a second later, the effect is gone, then you wouldn't really say that I had measured the electron's position or spin or momentum. You would only say that I'd measured something if later, maybe tomorrow, I still had a record (a memory, or a value written on paper, or online notes, or something) of the corresponding value. The point of [itex]lim_{t \rightarrow \infty} A(t)[/itex] is that the effect should still be present long after the cause (the electron passing by) is gone.

The way I would have said it is that a measurement happens when an interaction causes an irreversible change in the measuring device's state.

I would interpret it as A affects B, B affects A, A affects B etc. etc. until the system settles down in a time t, which need not be so long in our viewpoint.

Sorry @Erland I misunderstood your comment. @stevendaryl has a different take on "time going to infinity" which may be essentially equivalent to what I said, I'm not sure. Anyway, considering my answer, it's really a very standard approach in science and mathematics. For instance suppose we have a system in a lab and want to integrate some quantity over the whole. We could say the system stops within the boundaries of the apparatus, a few millimeters or meters. But a quantum particle can tunnel through, even beyond the walls of the lab. Of course that's usually negligible. But instead of hassling with defining those boundaries exactly, it's not unusual to integrate from - to + infinity. The quantity is going to be very close to zero within a short range, and vanishes at infinity, so such integral is equivalent to a finite one. But it's a lot more convenient, especially for theoretical work. Integrals are often easier to compute that way, than with finite limits. This sort of thing is ubiquitous in science. Going to infinity intuitively means, as far as necessary to cover everything non-negligible.