Theorems · Theorem · measure theory
MeasureTheory.lintegral_smul_measure
∀ {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {R : Type u_4} [inst : SMul R ENNReal]
[inst_1 : IsScalarTower R ENNReal ENNReal] (c : R) (f : α → ENNReal), ∫⁻ (a : α), f a ∂c • μ = c • ∫⁻ (a : α), f a ∂μ- Cited by
- 20 results in Mathlib
- Foundations
- Depth 196 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- SMulIsScalarTower
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites14
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coeproof · cited by 62,936
- MeasurableSpacestatement and proof · cited by 13,106
- MeasureTheory.Measurestatement and proof · cited by 10,939
- ENNRealstatement and proof · cited by 9,879
- IsScalarTowerstatement and proof · cited by 3,896
- iSupproof · cited by 2,415
- MeasureTheory.lintegralstatement · cited by 1,152
- MeasureTheory.SimpleFuncproof · cited by 411
- iSup_congr_Propproof · cited by 247
- MeasureTheory.SimpleFunc.lintegralproof · cited by 55
- iSup_subtype'proof · cited by 44
- MeasureTheory.lintegral_defproof · cited by 15
Cited by20
Results whose statement or proof uses this declaration.
- MeasureTheory.laverage_eqproof · cited by 12
- MeasureTheory.lintegral_countable'proof · cited by 4
- ProbabilityTheory.avgRisk_const_right'proof · cited by 3
- MeasureTheory.Measure.prod_smul_leftproof · cited by 3
- MeasureTheory.withDensity_smul_measureproof · cited by 3
- MeasureTheory.lintegral_singletonproof · cited by 2
- MeasureTheory.measure_le_setLAverage_posproof · cited by 2
- MeasureTheory.Measure.join_smulproof · cited by 2
- MeasureTheory.integral_of_ae_eq_zero_or_oneproof · cited by 2
- MeasureTheory.lintegral_pow_le_pow_lintegral_fderivproof · cited by 1
- MeasureTheory.FiniteMeasure.smul_testAgainstNN_applyproof · cited by 1
- MeasureTheory.lintegral_singleton'proof · cited by 0