Theorems · Theorem · measure theory

MeasureTheory.integrableOn_Ioi_comp_mul_left_iff

∀ {E : Type u_1} [inst : NormedAddCommGroup E] (f : ℝ → E) (c : ℝ) {a : ℝ},
  0 < a →
    (MeasureTheory.IntegrableOn (fun x => f (a * x)) (Set.Ioi c) MeasureTheory.volume ↔
      MeasureTheory.IntegrableOn f (Set.Ioi (a * c)) MeasureTheory.volume)

Defined in: Mathlib.MeasureTheory.Integral.IntegralEqImproper
Cited by: 5 results in Mathlib
Foundations: Depth 258 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: NormedAddCommGroup

Mathlib docs Source Loogle LeanSearch

Around this declaration

Dashed lines are statement dependencies; solid lines are citations in proofs.

Cites20

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

Setproof · cited by 53,352
Realstatement and proof · cited by 25,697
TopologicalSpaceproof · cited by 24,529
NormedAddCommGroupstatement and proof · cited by 15,752
MeasurableSpaceproof · cited by 13,106
MeasureTheory.Measureproof · cited by 10,939
mul_commproof · cited by 2,262
Set.Ioistatement and proof · cited by 1,463
LT.lt.ne'proof · cited by 1,417
MeasureTheory.Integrableproof · cited by 1,367
MeasureTheory.MeasureSpace.volumestatement and proof · cited by 1,323
Set.indicatorproof · cited by 723

Cited by5

Results whose statement or proof uses this declaration.

integrableOn_exp_mul_complex_Ioiproof · cited by 4
hasSum_mellinproof · cited by 1
integrableOn_rpow_mul_exp_neg_mul_rpowproof · cited by 1
MellinConvergent.comp_mul_leftproof · cited by 0
MeasureTheory.integrableOn_Ioi_comp_mul_right_iffproof · cited by 0