| import OAI.Analysis.CircleDomains.Modulus.PositiveCountableWeights | |
| namespace OAI | |
| /-! Finite positive costs on countable mark sets, from geometric weights | |
| of total mass at most one. -/ | |
| open scoped ENNReal | |
| noncomputable section | |
| namespace CircleDomainRigidity | |
| theorem exists_positive_finite_square_weights (S : Type*) [Countable S] : | |
| ∃ w : S → ℝ≥0∞, (∀ s, w s ≠ 0) ∧ (∀ s, w s ≠ ∞) ∧ | |
| (∑' s, w s) < ∞ ∧ (∑' s, w s ^ 2) < ∞ := by | |
| obtain ⟨w, hw, hsum⟩ := exists_positive_summable_weights S | |
| have hw1 : ∀ s, w s ≤ 1 := fun s => (ENNReal.le_tsum s).trans hsum | |
| refine ⟨w, fun s => (hw s).ne', fun s => ne_of_lt ((hw1 s).trans_lt (by simp)), | |
| hsum.trans_lt (by simp), ?_⟩ | |
| apply lt_of_le_of_lt (ENNReal.tsum_le_tsum ?_) (hsum.trans_lt (by simp)) | |
| intro s | |
| simpa only [pow_two, mul_one] using mul_le_mul' le_rfl (hw1 s) | |
| end CircleDomainRigidity | |
| end | |
| end OAI | |