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9eeb9c7 | 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 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 | import Mathlib.MeasureTheory.Function.L2Space
import Mathlib.MeasureTheory.Function.StronglyMeasurable.Lemmas
namespace OAI
noncomputable section
open MeasureTheory Filter
open scoped ENNReal Topology
namespace Invariant233.DirectIntegral
variable {X E F : Type*} [MeasurableSpace X] {μ : Measure X}
[NormedAddCommGroup E] [NormedSpace ℂ E]
[NormedAddCommGroup F] [NormedSpace ℂ F]
theorem apply_simple_aestronglyMeasurable
(A : X → E →L[ℂ] F)
(hm : ∀ v, AEStronglyMeasurable (fun x => A x v) μ) (s : SimpleFunc X E) :
AEStronglyMeasurable (fun x => A x (s x)) μ := by
classical
induction s using SimpleFunc.induction' with
| const v => exact hm v
| @pcw s t u hu hs ht =>
convert AEStronglyMeasurable.piecewise hu hs.restrict ht.restrict using 1
funext x
by_cases hx : x ∈ u <;> simp [SimpleFunc.coe_piecewise, hx]
theorem apply_aestronglyMeasurable
(A : X → E →L[ℂ] F)
(hm : ∀ v, AEStronglyMeasurable (fun x => A x v) μ)
{f : X → E} (hf : AEStronglyMeasurable f μ) :
AEStronglyMeasurable (fun x => A x (f x)) μ := by
apply aestronglyMeasurable_of_tendsto_ae atTop
(f := fun n x => A x (hf.stronglyMeasurable_mk.approx n x))
· intro n
exact apply_simple_aestronglyMeasurable A hm _
· filter_upwards [hf.ae_eq_mk] with x hx
rw [hx]
exact ((A x).continuous.tendsto _).comp (hf.stronglyMeasurable_mk.tendsto_approx x)
def fieldOperatorVec (A : X → E →L[ℂ] F) (C : ℝ)
(hA : ∀ x, ‖A x‖ ≤ C)
(hm : ∀ v, AEStronglyMeasurable (fun x => A x v) μ)
(f : Lp E 2 μ) : Lp F 2 μ :=
((Lp.memLp f).of_le_mul (apply_aestronglyMeasurable A hm (Lp.aestronglyMeasurable f))
(Filter.Eventually.of_forall fun x => (A x).le_of_opNorm_le (hA x) (f x))).toLp _
theorem fieldOperatorVec_ae (A : X → E →L[ℂ] F) (C : ℝ)
(hA : ∀ x, ‖A x‖ ≤ C)
(hm : ∀ v, AEStronglyMeasurable (fun x => A x v) μ)
(f : Lp E 2 μ) :
fieldOperatorVec A C hA hm f =ᵐ[μ] fun x => A x (f x) :=
MemLp.coeFn_toLp _
def fieldOperatorLinear (A : X → E →L[ℂ] F) (C : ℝ)
(hA : ∀ x, ‖A x‖ ≤ C)
(hm : ∀ v, AEStronglyMeasurable (fun x => A x v) μ) :
Lp E 2 μ →ₗ[ℂ] Lp F 2 μ where
toFun := fieldOperatorVec A C hA hm
map_add' f g := by
apply Lp.ext
filter_upwards [fieldOperatorVec_ae A C hA hm (f + g),
fieldOperatorVec_ae A C hA hm f, fieldOperatorVec_ae A C hA hm g,
Lp.coeFn_add f g,
Lp.coeFn_add (fieldOperatorVec A C hA hm f) (fieldOperatorVec A C hA hm g)]
with x hfg hf hg hsum hout
simp only [Pi.add_apply] at hsum hout
rw [hfg, hout, hf, hg, hsum, map_add]
map_smul' c f := by
change fieldOperatorVec A C hA hm (c • f) = c • fieldOperatorVec A C hA hm f
apply Lp.ext
filter_upwards [fieldOperatorVec_ae A C hA hm (c • f),
fieldOperatorVec_ae A C hA hm f, Lp.coeFn_smul c f,
Lp.coeFn_smul c (fieldOperatorVec A C hA hm f)] with x hcf hf hsmul hout
simp only [Pi.smul_apply] at hsmul hout
rw [hcf, hout, hf, hsmul, map_smul]
def fieldOperator (A : X → E →L[ℂ] F) (C : ℝ)
(hA : ∀ x, ‖A x‖ ≤ C)
(hm : ∀ v, AEStronglyMeasurable (fun x => A x v) μ) :
Lp E 2 μ →L[ℂ] Lp F 2 μ :=
(fieldOperatorLinear A C hA hm).mkContinuous C (by
intro f
apply Lp.norm_le_mul_norm_of_ae_le_mul
change ∀ᵐ x ∂μ, ‖fieldOperatorVec A C hA hm f x‖ ≤ C * ‖f x‖
filter_upwards [fieldOperatorVec_ae A C hA hm f] with x hx
rw [hx]
exact (A x).le_of_opNorm_le (hA x) (f x))
end Invariant233.DirectIntegral
end
end OAI
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