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feat(StandardBorel): BorelSpace instances for sum and sigma types #519

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30 changes: 21 additions & 9 deletions BrownianMotion/Auxiliary/StandardBorel.lean
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Expand Up @@ -4,23 +4,35 @@ public import Mathlib.MeasureTheory.Constructions.Polish.Basic

@[expose] public section

open Topology

variable {α β ι : Type*} {γ : ι → Type*} [Countable ι] [MeasurableSpace α] [MeasurableSpace β]
[∀ n, MeasurableSpace (γ n)]

instance BorelSpace.sum [TopologicalSpace α] [TopologicalSpace β] [BorelSpace α] [BorelSpace β] :
BorelSpace (α ⊕ β) := by
constructor
rw [Sum.instMeasurableSpace]
refine le_antisymm ?_ ?_
· sorry
· refine MeasurableSpace.generateFrom_le fun t ht ↦ ?_
simp only [isOpen_sum_iff, Set.mem_ofPred_eq] at ht
rw [measurableSet_sum_iff]
exact ⟨ht.1.measurableSet, ht.2.measurableSet⟩
refine ⟨le_antisymm ?_ ?_⟩
· borelize (α ⊕ β)
intro s hs
obtain ⟨hl, hr⟩ := measurableSet_sum_iff.mp hs
rw [← Set.image_preimage_inl_union_image_preimage_inr s]
refine MeasurableSet.union ?_ ?_
· exact IsOpenEmbedding.inl.measurableEmbedding.measurableSet_image' hl
· exact IsOpenEmbedding.inr.measurableEmbedding.measurableSet_image' hr
· refine MeasurableSpace.generateFrom_le fun t ht ↦ measurableSet_sum_iff.mpr ?_
exact (isOpen_sum_iff.mp ht).imp IsOpen.measurableSet IsOpen.measurableSet

instance BorelSpace.sigma [∀ n, TopologicalSpace (γ n)] [∀ n, BorelSpace (γ n)] :
BorelSpace ((n : ι) × γ n) := by
sorry
refine ⟨le_antisymm ?_ ?_⟩
· borelize ((n : ι) × γ n)
intro s hs
rw [← Set.iUnion_image_preimage_sigma_mk_eq_self s]
refine MeasurableSet.iUnion fun n ↦ MeasurableEmbedding.measurableSet_image' ?_ ?_
· exact IsOpenEmbedding.sigmaMk.measurableEmbedding
· exact MeasurableSpace.measurableSet_iInf.mp hs n
· refine MeasurableSpace.generateFrom_le fun t ht ↦ ?_
exact MeasurableSpace.measurableSet_iInf.mpr fun n ↦ (isOpen_sigma_iff.mp ht n).measurableSet

/-- A sum of two standard Borel spaces is standard Borel. -/
instance StandardBorelSpace.sum [StandardBorelSpace α] [StandardBorelSpace β] :
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