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feat: the numeraires of the sample sizes need not form an e-process, and two-point nulls on finite spaces #437

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feat: the numeraires of the sample sizes need not form an e-process, and two-point nulls on finite spaces #437

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feat: the numeraires of the sample sizes need not form an e-process, …
RemyDegenne Nov 15, 2026
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refactor: expose the i.i.d. and piLE helpers, split off the Bernoulli…
RemyDegenne Nov 15, 2026
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10 changes: 10 additions & 0 deletions LMLExplore/EValues/Numeraire.lean
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Expand Up @@ -70,6 +70,8 @@ finite.
* `ProbabilityTheory.isNumeraire_rnDeriv`: for `Q ≪ P`, the likelihood ratio `Q.rnDeriv P` is the
numeraire of `{P}` [RW, Theorem 3.20]; its expected logarithm is `klDiv Q P` by Statlib's
`maxUtility_singleton_eq_klDiv`.
* `ProbabilityTheory.IsNumeraire.eq_of_countable`: on a countable space where `Q` charges every
point, numeraires are unique everywhere.

## References

Expand Down Expand Up @@ -413,4 +415,12 @@ theorem ripr_singleton_eq [IsProbabilityMeasure Q] [IsProbabilityMeasure P] (hQP
ripr_eq_of_isEVar_rnDeriv (by rintro _ rfl; infer_instance)
(subset_effectiveSet (by simp) rfl) hQP hPQ (isNumeraire_rnDeriv hQP).isEVar

/-! ### Uniqueness on a countable space -/

/-- On a countable space where `Q` charges every point, two numeraires of the same null under `Q`
are equal everywhere, not only `Q`-almost surely. -/
theorem IsNumeraire.eq_of_countable [Countable Ω] {Y : Ω → ℝ≥0∞} (hX : IsNumeraire X S Q)
(hY : IsNumeraire Y S Q) (hQ : ∀ ω, Q {ω} ≠ 0) : X = Y :=
funext fun ω ↦ ae_iff_of_countable.1 (hX.ae_unique hY) ω (hQ ω)

end ProbabilityTheory
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