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9b6e014 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 | import OAI.Probability.ContinuumTransition.PhysicalPressure
namespace OAI
universe u_1 u_2
section
open scoped BigOperators
open MeasureTheory Set
namespace ContinuumTemperature
section
variable {X : Type u_1} [MeasurableSpace X] {ι : Type u_2} [Fintype ι] [DecidableEq ι]
omit [MeasurableSpace X] [Fintype ι] in
theorem binaryRegion_subset_union (A B : Set X) (s : Finset ι) :
binaryRegion A B s ⊆ {x : ι → X | ∀ i, x i ∈ A ∪ B} := by
intro x hx i
by_cases hi : i ∈ s
· exact Or.inl (by simpa only [ite_eq_left hi] using hx i)
· exact Or.inr (by simpa only [ite_eq_right hi] using hx i)
omit [MeasurableSpace X] in
theorem iUnion_binaryRegion (A B : Set X) :
(⋃ s : Finset ι, binaryRegion A B s) = {x : ι → X | ∀ i, x i ∈ A ∪ B} := by
classical
ext x
constructor
· intro hx
obtain ⟨s, hs⟩ := mem_iUnion.mp hx
exact binaryRegion_subset_union A B s hs
· intro hx
refine mem_iUnion.mpr ⟨Finset.univ.filter (fun i => x i ∈ A), ?_⟩
intro i
by_cases hi : x i ∈ A
· simpa only [Finset.mem_filter, Finset.mem_univ, true_and, ite_eq_left hi] using hi
· simpa only [Finset.mem_filter, Finset.mem_univ, true_and, ite_eq_right hi] using
(hx i).resolve_left hi
theorem integral_binary_assignments_eq (μ : Measure X) [SigmaFinite μ]
{A B : Set X} (hA : MeasurableSet A) (hB : MeasurableSet B) (hd : Disjoint A B)
{F : (ι → X) → ℝ}
(hF : IntegrableOn F {x : ι → X | ∀ i, x i ∈ A ∪ B} (Measure.pi (fun _ : ι => μ))) :
(∫ x in {x : ι → X | ∀ i, x i ∈ A ∪ B}, F x ∂(Measure.pi (fun _ : ι => μ))) =
∑ s : Finset ι, ∫ x in binaryRegion A B s, F x ∂(Measure.pi (fun _ : ι => μ)) := by
have hh := integral_biUnion_finset (Finset.univ : Finset (Finset ι))
(fun s _ => measurableSet_binaryRegion hA hB s)
(fun _ _ _ _ hne => binaryRegion_pairwiseDisjoint hd hne)
(fun s _ => hF.mono_set (binaryRegion_subset_union A B s))
simpa only [Finset.mem_univ, iUnion_true, iUnion_binaryRegion] using hh
end
end ContinuumTemperature
end
end OAI
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