feat: sups and limsups are measurable in conditionally complete linea…

…r orders (#6979)

Currently, we only have that sups and limsups are measurable in complete linear orders, which excludes the main case of the real line. With more complicated proofs, these measurability results can be extended to all conditionally complete linear orders, without any further assumption in the statements.

Mathlib/Data/Set/Lattice.lean

@@ -2168,9 +2168,18 @@ end Disjoint
 
 /-! ### Intervals -/
 
-
 namespace Set
 
+lemma nonempty_iInter_Iic_iff [Preorder α] {f : ι → α} :
+    (⋂ i, Iic (f i)).Nonempty ↔ BddBelow (range f) := by
+  have : (⋂ (i : ι), Iic (f i)) = lowerBounds (range f) := by
+    ext c; simp [lowerBounds]
+  simp [this, BddBelow]
+
+lemma nonempty_iInter_Ici_iff [Preorder α] {f : ι → α} :
+    (⋂ i, Ici (f i)).Nonempty ↔ BddAbove (range f) :=
+  nonempty_iInter_Iic_iff (α := αᵒᵈ)
+
 variable [CompleteLattice α]
 
 theorem Ici_iSup (f : ι → α) : Ici (⨆ i, f i) = ⋂ i, Ici (f i) :=