Yes, this is called the Finkelstein-Hartle theorem (D. Finkelstein, Transactions of the New York Academy of Sciences 25, 621 (1963); J. B. Hartle, Am. J. Phys. 36, 704 (1968)).
This theorem is the basis for constructing a limit operator for the relative frequency when there are infinitely many independent repetitions of a measurement, and showing that the product wave-function is an exact eigenstate of the relative frequency operator. Unfortunately, it seems that Hartle’s construction of the frequency operator wasn’t quite right, and needed to be generalized. (E. Farhi, J. Goldstone, and S. Gutmann, Ann. Phys. 192, 368 (1989)).
Even so, the critics are still picky about the construction. There is a line of criticism that infinite frequency operators can be constructed arbitrarily as functions over Hilbert space, and unless you already know the Born rule, you won’t know how to construct one sensibly (so that the Hartle derivation is circular). However this seems unfair, because if you want the relative frequency operator to obey the Kolmogorov axioms of probability then it has to coincide with the Born rule, something which is another long-standing result called Gleason’s theorem. (The squared modulus of the amplitude is the only function of the measure which follows the axioms of probability.) Hence the full derivation is:
1) (Postulate) If the wavefunction is in an eigenstate of a measurement operator, then the measurement will with certainty have the corresponding eigenvalue.
2) (Postulate) Probability is relative frequency over infinitely many independent repetitions.
3) (Postulate) Relative frequency follows the Kolmogorov axioms of probability.
4) (Gleason’s theorem) Relative frequency must converge to the Born rule (squared modulus of amplitude) over infinitely many repetitions, or it won’t be able to follow the Kolmogorov axioms.
5) (Hartle’s theorem, as strengthened by Farhi et al) There is a unique definition of the relative frequency operator over infinite repetitions, and such that the infinite product state is an eigenstate of the relative frequency operator.
6) (Conclusion) The relative frequency over infinitely many measurements is with certainty the Born probability.
Yes, this is called the Finkelstein-Hartle theorem (D. Finkelstein, Transactions of the New York Academy of Sciences 25, 621 (1963); J. B. Hartle, Am. J. Phys. 36, 704 (1968)).
This theorem is the basis for constructing a limit operator for the relative frequency when there are infinitely many independent repetitions of a measurement, and showing that the product wave-function is an exact eigenstate of the relative frequency operator. Unfortunately, it seems that Hartle’s construction of the frequency operator wasn’t quite right, and needed to be generalized. (E. Farhi, J. Goldstone, and S. Gutmann, Ann. Phys. 192, 368 (1989)).
Even so, the critics are still picky about the construction. There is a line of criticism that infinite frequency operators can be constructed arbitrarily as functions over Hilbert space, and unless you already know the Born rule, you won’t know how to construct one sensibly (so that the Hartle derivation is circular). However this seems unfair, because if you want the relative frequency operator to obey the Kolmogorov axioms of probability then it has to coincide with the Born rule, something which is another long-standing result called Gleason’s theorem. (The squared modulus of the amplitude is the only function of the measure which follows the axioms of probability.) Hence the full derivation is:
1) (Postulate) If the wavefunction is in an eigenstate of a measurement operator, then the measurement will with certainty have the corresponding eigenvalue.
2) (Postulate) Probability is relative frequency over infinitely many independent repetitions.
3) (Postulate) Relative frequency follows the Kolmogorov axioms of probability.
4) (Gleason’s theorem) Relative frequency must converge to the Born rule (squared modulus of amplitude) over infinitely many repetitions, or it won’t be able to follow the Kolmogorov axioms.
5) (Hartle’s theorem, as strengthened by Farhi et al) There is a unique definition of the relative frequency operator over infinite repetitions, and such that the infinite product state is an eigenstate of the relative frequency operator.
6) (Conclusion) The relative frequency over infinitely many measurements is with certainty the Born probability.
It seem pretty clean to me.