Theorems · Theorem · probability
ProbabilityTheory.IsGaussian.integral_dual_conv_map_neg_eq_zero
∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E] [inst_2 : MeasurableSpace E] [BorelSpace E]
{μ : MeasureTheory.Measure E} [ProbabilityTheory.IsGaussian μ] [SecondCountableTopology E] (L : StrongDual ℝ E),
∫ (x : E), L x ∂μ.conv (MeasureTheory.Measure.map (⇑(ContinuousLinearEquiv.neg ℝ)) μ) = 0The convolution of a Gaussian measure μ and its map by x ↦ -x is centered.
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 308 from the axioms · uses propext, Classical.choice, Quot.sound
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Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
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- Realstatement and proof · cited by 25,697
- RingHom.idstatement · cited by 18,349
- NormedAddCommGroupstatement and proof · cited by 15,752
- MeasurableSpacestatement and proof · cited by 13,106
- NormedSpacestatement and proof · cited by 12,499
- MeasureTheory.Measurestatement and proof · cited by 10,939
- one_mulproof · cited by 2,841
- MeasureTheory.integralstatement and proof · cited by 1,779
- BorelSpacestatement and proof · cited by 1,602
- map_addproof · cited by 964
- MeasureTheory.Measure.mapstatement and proof · cited by 858
Cited by1
Results whose statement or proof uses this declaration.
- ProbabilityTheory.IsGaussian.exists_integrable_exp_sqproof · cited by 1