Theorems · Theorem · measure theory
MeasureTheory.average_eq
∀ {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E]
(μ : MeasureTheory.Measure α) (f : α → E), ⨍ (x : α), f x ∂μ = (μ.real Set.univ)⁻¹ • ∫ (x : α), f x ∂μ- Defined in
- Mathlib.MeasureTheory.Integral.Average
- Cited by
- 10 results in Mathlib
- Foundations
- Depth 253 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites15
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coeproof · cited by 62,936
- 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
- Set.univstatement and proof · cited by 3,945
- MeasureTheory.integralstatement and proof · cited by 1,779
- ENNReal.toRealproof · cited by 859
- MeasureTheory.Measure.realstatement and proof · cited by 530
- MeasureTheory.averagestatement · cited by 87
- MeasureTheory.measureReal_defproof · cited by 68
Cited by10
Results whose statement or proof uses this declaration.
- MeasureTheory.setAverage_eqproof · cited by 5
- MeasureTheory.measure_smul_averageproof · cited by 4
- MeasureTheory.toReal_laverageproof · cited by 3
- MeasureTheory.average_congrproof · cited by 2
- MeasureTheory.average_add_measureproof · cited by 1
- MeasureTheory.tendsto_integral_smul_of_tendsto_average_norm_subproof · cited by 1
- ae_eq_const_or_norm_average_lt_of_norm_le_constproof · cited by 1
- ae_eq_const_or_norm_integral_lt_of_norm_le_constproof · cited by 1
- MeasureTheory.ofReal_averageproof · cited by 1
- MeasureTheory.setAverage_congr_funproof · cited by 0