Plain-English statement
- Remainder measurability transported from the normalized Brownian coordinate law. This cycle-194 lower_2 bridge discharges the downstream `hRemainderMeas` shape once the source correspondence has supplied measurability under the normalized scalar-coordinate law. The law transport uses only the already compiled standard-Gaussian coordinate-law bridge and the variance packaging bridge; domination and integrability remain separate leaves.
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Lean statement
theorem selectedWeakTestRemainderMeasOfStdGaussianVectorLaw
{Test E : Type*} [NormedAddCommGroup E] [InnerProductSpace Real E]
[FiniteDimensional Real E] [MeasurableSpace E] [BorelSpace E]
(variance : Test → E → Fin (Module.finrank Real E) → NNReal)
(normalizedVectorLaw : Test → E → MeasureTheory.Measure E)
(normalizedCoordinateLaw :
Test → E → Fin (Module.finrank Real E) → MeasureTheory.Measure Real)
(normalizedRemainder :
Test → E → Fin (Module.finrank Real E) → Real → Real)
(testRegular : Prop)
(hNormalizedRemainderMeas :
testRegular →
∀ φ x i,
MeasureTheory.AEStronglyMeasurable
(normalizedRemainder φ x i)
(normalizedCoordinateLaw φ x i))
(hNormalizedVectorLaw :
testRegular →
∀ φ x, normalizedVectorLaw φ x = ProbabilityTheory.stdGaussian E)
(hCoordinateLawDef :
testRegular →
∀ φ x i,
normalizedCoordinateLaw φ x i =
(normalizedVectorLaw φ x).map
(fun y : E => inner Real ((stdOrthonormalBasis Real E) i) y))
(hVarianceDef :
testRegular →
∀ φ x i,
(variance φ x i : Real) =
ProbabilityTheory.variance (id : Real → Real)
(normalizedCoordinateLaw φ x i)) :
testRegular →
∀ φ x i,
MeasureTheory.AEStronglyMeasurable
(fun z : Real => normalizedRemainder φ x i z)
(ProbabilityTheory.gaussianReal (0 : Real) (variance φ x i)) := by
intro htests φ x i
have hNormalizedCoordinateLaw :
testRegular →
∀ φ x i,
normalizedCoordinateLaw φ x i =
ProbabilityTheory.gaussianReal (0 : Real) (1 : NNReal) :=
selectedWeakTestNormalizedCoordinateLawOfStdGaussianVectorLaw
normalizedVectorLaw normalizedCoordinateLaw testRegular
hNormalizedVectorLaw hCoordinateLawDef
have hVarianceOne :
testRegular →
∀ φ x i, variance φ x i = (1 : NNReal) :=
selectedWeakTestNormalizedVarianceDefOfGaussianRealUnitLaw
variance normalizedCoordinateLaw testRegular hVarianceDef
hNormalizedCoordinateLaw
simpa [hNormalizedCoordinateLaw htests φ x i,
hVarianceOne htests φ x i]
using hNormalizedRemainderMeas htests φ x i
/-- Remainder domination transported from the normalized Brownian coordinate law.
This cycle-194 bridge narrows the downstream `hRemainderBound` leaf to the
same source-side normalized scalar-coordinate law used for the remainder
measurability bridge. It only rewrites the ambient measure using the compiled
standard-Gaussian coordinate-law bridge and the variance packaging bridge;
integrability of the dominating bound remains a separate leaf.
-/Proof architecture
discharge hRemainderMeas in the active Brownian/Ito Taylor moment backend
Lean proof walkthrough
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Why the statement has this shape
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Hidden assumptions and non-claims
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