Theorems · Theorem · measure theory
MeasureTheory.average_eq_integral
∀ {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E]
(μ : MeasureTheory.Measure α) [MeasureTheory.IsProbabilityMeasure μ] (f : α → E),
⨍ (x : α), f x ∂μ = ∫ (x : α), f x ∂μ- Defined in
- Mathlib.MeasureTheory.Integral.Average
- Cited by
- 11 results in Mathlib
- Foundations
- Depth 252 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites12
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
- 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
- ENNRealproof · cited by 9,879
- MeasureTheory.integralstatement and proof · cited by 1,779
- one_smulproof · cited by 1,374
- MeasureTheory.IsProbabilityMeasurestatement and proof · cited by 392
- inv_oneproof · cited by 301
- MeasureTheory.IsProbabilityMeasure.measure_univproof · cited by 135
- MeasureTheory.averagestatement · cited by 87
Cited by11
Results whose statement or proof uses this declaration.
- ProbabilityTheory.condVar_botproof · cited by 1
- MeasureTheory.le_integral_rnDeriv_of_acproof · cited by 1
- ConvexOn.apply_rnDeriv_ae_le_integralproof · cited by 1
- MeasureTheory.measure_le_integral_posproof · cited by 0
- MeasureTheory.exists_integral_leproof · cited by 0
- MeasureTheory.measure_integral_le_posproof · cited by 0
- MeasureTheory.exists_le_integralproof · cited by 0
- ConvexOn.map_integral_leproof · cited by 0
- MeasureTheory.exists_notMem_null_integral_leproof · cited by 0
- ConcaveOn.le_map_integralproof · cited by 0
- MeasureTheory.exists_notMem_null_le_integralproof · cited by 0