English
If a left-invariant measure μ is sigma-finite and finite on a compact with nonempty interior, then μ is regular.
Русский
Если слева инвариантная мера μ сигма-конечна и конечна на компактном с непустымInterior, то μ регулярна.
LaTeX
$$Regular μ follows from Regular.smul with μ(K) ≠ ∞ and finite on K$$
Lean4
/-- To show that an invariant σ-finite measure is regular it is sufficient to show that it is finite
on some compact set with non-empty interior. -/
@[to_additive /-- To show that an invariant σ-finite measure is regular it is sufficient to show that it is
finite on some compact set with non-empty interior. -/
]
theorem regular_of_isMulLeftInvariant {μ : Measure G} [SigmaFinite μ] [IsMulLeftInvariant μ] {K : Set G}
(hK : IsCompact K) (h2K : (interior K).Nonempty) (hμK : μ K ≠ ∞) : Regular μ := by
rw [haarMeasure_unique μ ⟨⟨K, hK⟩, h2K⟩]; exact Regular.smul hμK
