feat(measure_theory/borel_space): a preconnected set is measurable (#…

…8044)

In a conditionally complete linear order equipped with the order topology and the corresponding borel σ-algebra, any preconnected set is measurable.

src/data/set/finite.lean

@@ -3,7 +3,7 @@ Copyright (c) 2017 Johannes Hölzl. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Johannes Hölzl, Mario Carneiro
 -/
-import data.fintype.basic
+import data.finset.sort
 
 /-!
 # Finite sets
@@ -810,3 +810,28 @@ lemma card_compl_eq_card_compl (h : fintype.card {x // p x} = fintype.card {x //
 by simp only [card_subtype_compl, h]
 
 end fintype
+
+/--
+If a set `s` does not contain any elements between any pair of elements `x, z ∈ s` with `x ≤ z`
+(i.e if given `x, y, z ∈ s` such that `x ≤ y ≤ z`, then `y` is either `x` or `z`), then `s` is
+finite.
+-/
+lemma set.finite_of_forall_between_eq_endpoints {α : Type*} [linear_order α] (s : set α)
+  (h : ∀ (x ∈ s) (y ∈ s) (z ∈ s), x ≤ y → y ≤ z → x = y ∨ y = z) :
+  set.finite s :=
+begin
+  by_contra hinf,
+  change s.infinite at hinf,
+  rcases hinf.exists_subset_card_eq 3 with ⟨t, hts, ht⟩,
+  let f := t.order_iso_of_fin ht,
+  let x := f 0,
+  let y := f 1,
+  let z := f 2,
+  have := h x (hts x.2) y (hts y.2) z (hts z.2)
+    (f.monotone $ by dec_trivial) (f.monotone $ by dec_trivial),
+  have key₁ : (0 : fin 3) ≠ 1 := by dec_trivial,
+  have key₂ : (1 : fin 3) ≠ 2 := by dec_trivial,
+  cases this,
+  { dsimp only [x, y] at this, exact key₁ (f.injective $ subtype.coe_injective this) },
+  { dsimp only [y, z] at this, exact key₂ (f.injective $ subtype.coe_injective this) }
+end

src/measure_theory/borel_space.lean

@@ -351,17 +351,36 @@ instance nhds_within_Iio_is_measurably_generated :
   (𝓝[Iio b] a).is_measurably_generated :=
 measurable_set_Iio.nhds_within_is_measurably_generated _
 
-variables [second_countable_topology α]
-
 @[measurability]
-lemma measurable_set_lt' : measurable_set {p : α × α | p.1 < p.2} :=
+lemma measurable_set_lt' [second_countable_topology α] : measurable_set {p : α × α | p.1 < p.2} :=
 (is_open_lt continuous_fst continuous_snd).measurable_set
 
 @[measurability]
-lemma measurable_set_lt {f g : δ → α} (hf : measurable f) (hg : measurable g) :
-  measurable_set {a | f a < g a} :=
+lemma measurable_set_lt [second_countable_topology α] {f g : δ → α} (hf : measurable f)
+  (hg : measurable g) : measurable_set {a | f a < g a} :=
 hf.prod_mk hg measurable_set_lt'
 
+lemma set.ord_connected.measurable_set (h : ord_connected s) : measurable_set s :=
+begin
+  let u := ⋃ (x ∈ s) (y ∈ s), Ioo x y,
+  have huopen : is_open u := is_open_bUnion (λ x hx, is_open_bUnion (λ y hy, is_open_Ioo)),
+  have humeas : measurable_set u := huopen.measurable_set,
+  have hfinite : (s \ u).finite,
+  { refine set.finite_of_forall_between_eq_endpoints (s \ u) (λ x hx y hy z hz hxy hyz, _),
+    by_contra h,
+    push_neg at h,
+    exact hy.2 (mem_bUnion_iff.mpr ⟨x, hx.1,
+      mem_bUnion_iff.mpr ⟨z, hz.1, lt_of_le_of_ne hxy h.1, lt_of_le_of_ne hyz h.2⟩⟩) },
+  have : u ⊆ s :=
+    bUnion_subset (λ x hx, bUnion_subset (λ y hy, Ioo_subset_Icc_self.trans (h.out hx hy))),
+  rw ← union_diff_cancel this,
+  exact humeas.union hfinite.measurable_set
+end
+
+lemma is_preconnected.measurable_set
+  (h : is_preconnected s) : measurable_set s :=
+h.ord_connected.measurable_set
+
 end linear_order
 
 section linear_order

src/topology/algebra/ordered/basic.lean

@@ -679,6 +679,10 @@ lemma filter.tendsto.min {b : filter β} {a₁ a₂ : α} (hf : tendsto f b (
   tendsto (λb, min (f b) (g b)) b (𝓝 (min a₁ a₂)) :=
 (continuous_min.tendsto (a₁, a₂)).comp (hf.prod_mk_nhds hg)
 
+lemma is_preconnected.ord_connected {s : set α} (h : is_preconnected s) :
+  ord_connected s :=
+⟨λ x hx y hy, h.Icc_subset hx hy⟩
+
 end linear_order
 
 end order_closed_topology
@@ -2688,14 +2692,14 @@ end
 
 lemma is_preconnected_interval : is_preconnected (interval a b) := is_preconnected_Icc
 
+lemma set.ord_connected.is_preconnected {s : set α} (h : s.ord_connected) :
+  is_preconnected s :=
+is_preconnected_of_forall_pair $ λ x y hx hy, ⟨interval x y, h.interval_subset hx hy,
+  left_mem_interval, right_mem_interval, is_preconnected_interval⟩
+
 lemma is_preconnected_iff_ord_connected {s : set α} :
   is_preconnected s ↔ ord_connected s :=
-⟨λ h, ⟨λ x hx y hy, h.Icc_subset hx hy⟩, λ h, is_preconnected_of_forall_pair $ λ x y hx hy,
-  ⟨interval x y, h.interval_subset hx hy, left_mem_interval, right_mem_interval,
-    is_preconnected_interval⟩⟩
-
-alias is_preconnected_iff_ord_connected ↔
-  is_preconnected.ord_connected set.ord_connected.is_preconnected
+⟨is_preconnected.ord_connected, set.ord_connected.is_preconnected⟩
 
 lemma is_preconnected_Ici : is_preconnected (Ici a) := ord_connected_Ici.is_preconnected
 lemma is_preconnected_Iic : is_preconnected (Iic a) := ord_connected_Iic.is_preconnected