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I've quite often heard the claim the many worlds interpretation of quantum mechanics denies the measurement (there is no such thing as the measurement).

Many Worlds Interpretation which basically states that the universe as a whole develops like an unobserved quantum system, and any observation effects ("collapse of the wave function") are illusions which are caused by the observer getting entangled with the observed system, which effectively causes a split of his world into many worlds, one for each measurement outcome.

But the question still remains why that subset of measurement outcomes? For example, if I take an electron microscope and measure the position of an electron there is no world in which I measure the momentum of the electron. So even the many-worlder must concede there is something like the measurement.

What am I missing?

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It does not deny the "measurement", it denies the collapse of a quantum state to a single measured outcome, because each possible outcome is realized in a separate branch. The privileged basis of possible outcomes is determined by decoherence, a momentum measurement is not in it on your scenario.

– Conifold – 2020-07-23T10:52:29.1332

I think this question is perhaps a small mis-reading. What you have written is consistent with my understanding of the matter with the exception of this the following: “the many worlds interpretation of quantum mechanics denies the measurement (there is no such thing as the measurement).”

I have never encountered this claim. What I have repeatedly encountered is that the many worlds interpretation denies the measurement

problem. Which means narrowly a denial of there being wave-fn collapse; not measurement, in the general sense.