Theorems · Theorem · probability

ProbabilityTheory.HasGaussianLaw.map_eq_gaussianReal

∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {P : MeasureTheory.Measure Ω} {X : Ω → ℝ},
  ProbabilityTheory.HasGaussianLaw X P →
    MeasureTheory.Measure.map X P =
      ProbabilityTheory.gaussianReal (∫ (x : Ω), X x ∂P) (ProbabilityTheory.variance X P).toNNReal

Defined in: Mathlib.Probability.Distributions.Gaussian.HasGaussianLaw.Basic
Cited by: 2 results in Mathlib
Foundations: Depth 295 from the axioms · uses propext, Classical.choice, Quot.sound

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Cites16

Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.

Realstatement and proof · cited by 25,697
MeasurableSpacestatement and proof · cited by 13,106
MeasureTheory.Measurestatement and proof · cited by 10,939
MeasureTheory.integralstatement and proof · cited by 1,779
MeasureTheory.Measure.mapstatement and proof · cited by 858
Real.toNNRealstatement and proof · cited by 267
ProbabilityTheory.variancestatement and proof · cited by 104
ProbabilityTheory.gaussianRealstatement and proof · cited by 77
MeasureTheory.integral_mapproof · cited by 67
ProbabilityTheory.HasGaussianLawstatement and proof · cited by 67
aemeasurable_idproof · cited by 66
aestronglyMeasurable_idproof · cited by 35

Cited by2

Results whose statement or proof uses this declaration.

ProbabilityTheory.BrownianReal.measurePreserving_eval_projectiveFamilyproof · cited by 1
ProbabilityTheory.BrownianReal.measurePreserving_eval_sub_eval_projectiveFamilyproof · cited by 1