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Measurability for quantifier-alternating functions defined by trigonometric polynomials

Establish whether ZFC proves that for every trigonometric polynomial g, any function obtained by applying an arbitrary finite sequence of suprema over R, followed by an arbitrary finite sequence of infima over R, followed again by an arbitrary finite sequence of suprema over R, and finally an arbitrary finite sequence of infima over R to g (i.e., any function of the form inf_R^* sup_R^* inf_R^* sup_R^* g) is Lebesgue measurable.

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Background

Using identities involving sine, the paper shows how to simulate quantification over integers by quantification over reals, yielding independence results for measurability with trigonometric polynomials but with an additional alternation of quantifiers. The author notes that it is unclear if this extra alternation is essential and formulates a concrete question about measurability at the level of four alternating quantifier blocks.

Existing positive results (e.g., for Borel/o-minimal/local o-minimal settings) and large-cardinal-based theorems do not directly settle this precise trigonometric-polynomial case within ZFC.

References

The fact that something like this should be possible follows from \cref{fact:sine-sin}, but it is unclear if the extra alternation of quantifiers is necessary. Does ZFC\ prove that for every trigonometric polynomial $g$, any function of the form $\inf_R\ast\sup_R\ast \inf_R\ast \sup_R\ast g$ is Lebesgue measurable?

— Any function I can actually write down is measurable, right? (2501.02693 - Hanson, 6 Jan 2025) in Section 6 (Measurability by inspection), after the table and before the appendix; Question (quest:trig)