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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
    Jul 23, 2020 at 10:52
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    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.
    – znr
    Jul 23, 2020 at 17:32

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MWI doesn't deny the existence of measurements. What it does is deny that any special postulates or physics are needed to explain measurement. Systems just evolve according to the Schrodinger equation, whether or not they are being measured; measurement is a socially defined term for a subset of system evolutions contrived by humans to gain information about systems being measured.

Under textbook QM systems evolve according to the Schrodinger equation except when measured, when they collapse to one of their eigenstates with a probability given by the Born rule. This is clearly very unsatisfactory as a vague high level concept like 'measurement' cannot be defined or distinguished in terms of fundamental physics.

For MWI this collapse does not occur; when I do a measurement I do indeed see only one eigenstate with a probability given by the Born rule, but that because 'I' am one 'I' on one branch and there are other 'I's on other branches seeing the other eigenstates. The most interesting open questions (for me anyway) around the philosophical aspects of MWI are around how we make sense of this fundamentally indexical notion of 'worlds', as well as explicating how probabilities work here.

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If you take an electron microscope and measure the position of an electron, quantum mechanics predicts a range of possible positions, governed by its wave function and the uncertainty principle. In the many-worlds interpretation, each such possible position splits off its own universe in which it is the actual position. When you measure it in that particular position, all you are doing is discovering which of those many split worlds you have followed. There are now many other parallel yous who measured different positions.

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