| import OAI.Analysis.CircleDomains.Probability.LocalNullImage |
| import Mathlib.Topology.Path |
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|
| namespace OAI |
|
|
| /-! |
| # From simple-arc payment to null trace images |
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| Distinct points of a connected open subset of a locally connected metric |
| continuum can be joined there by a simple arc. The arc-existence hypothesis |
| is the local arcwise-connectedness theorem used by Lorimer (1981), |
| *Locally compact Hjelmslev planes and rings*, proof of Lemma 3.2, |
| pp. 1002–1003, displayed |
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| [TopologicalSpace X] : Prop := |
| ∀ U : Set X, IsOpen U → IsConnected U → |
| ∀ x ∈ U, ∀ y ∈ U, x ≠ y → |
| ∃ γ : Path x y, Function.Injective γ ∧ Set.range γ ⊆ U |
|
|
| /-- The null-image criterion with the actual simple-arc payment premise. |
| The input only supplies arcs; no oscillation or null-image conclusion |
| is included in it. -/ |
| theorem null_image_of_simple_arc_payment |
| {X : Type*} [MetricSpace X] [SecondCountableTopology X] [LocallyConnectedSpace X] |
| [MeasurableSpace X] [BorelSpace X] {μ : Measure X} [IsFiniteMeasure μ] |
| [μ.OuterRegular] (harcs : HasSimpleArcsInConnectedOpens X) |
| {F V : Set X} (hV : IsOpen V) (hF : μ (F ∩ V) = 0) |
| (v : X → ℝ) {A : ℝ} (hA : 0 ≤ A) |
| (hpay : ∀ x ∈ F ∩ V, ∀ y ∈ F ∩ V, |
| ∀ γ : Path x y, Function.Injective γ → range γ ⊆ V → |
| |v x - v y| ≤ A * μ.real (range γ)) : |
| volume (v '' (F ∩ V)) = 0 := by |
| apply null_image_inter_open_of_connected_open_oscillation hV hF v hA |
| intro U hUV hU hconn x hx y hy |
| by_cases hxy : x = y |
| · subst y |
| simpa only [sub_self, abs_zero] using |
| mul_nonneg hA (measureReal_nonneg (μ := μ) (s := U)) |
| obtain ⟨γ, hγ, hγU⟩ := harcs U hU hconn x hx.2 y hy.2 hxy |
| exact (hpay x ⟨hx.1, hUV hx.2⟩ y ⟨hy.1, hUV hy.2⟩ γ hγ |
| (hγU.trans hUV)).trans |
| (mul_le_mul_of_nonneg_left (measureReal_mono hγU) hA) |
|
|
| end CircleDomainRigidity |
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| end OAI |
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