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Correctness of Portnoy (2022) Lusin-based proof

Ascertain whether the proof in Portnoy (2022, “Linearity of Unbiased Linear Models Estimators,” The American Statistician 76, 372–375) that relies on Lusin’s Theorem to establish Lebesgue almost-everywhere linearity is correct.

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Background

The authors clarify that Portnoy (2022) claims only Lebesgue almost-everywhere linearity (not full linearity) and that the argument relies on Lusin’s Theorem. They explicitly state that they have not verified the correctness of that proof.

This leaves an unresolved question regarding the validity of Portnoy’s argument as written.

References

The proof in Portnoy (2022) rests on Lusin's Theorem and inspection shows that it only delivers Lebesgue a.e. linearity. Whether this proof is actually correct, we have not checked.

— Comments on B. Hansen's Reply to "A Comment on: `A Modern Gauss-Markov Theorem'", and Some Related Discussion (2406.03971 - Pötscher, 6 Jun 2024) in Section 2 (Comments on Portnoy (2023)), p. 3