Theorems · Theorem · measure theory

MeasureTheory.integral_Ioc_eq_integral_Ioo

∀ {X : Type u_1} {E : Type u_3} {mX : MeasurableSpace X} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E]
  {f : X → E} {μ : MeasureTheory.Measure X} [inst_2 : PartialOrder X] {x y : X} [MeasureTheory.NullSingletonClass μ],
  ∫ (t : X) in Set.Ioc x y, f t ∂μ = ∫ (t : X) in Set.Ioo x y, f t ∂μ

Defined in: Mathlib.MeasureTheory.Integral.Bochner.Set
Cited by: 6 results in Mathlib
Foundations: Depth 252 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: NormedAddCommGroup NormedSpace PartialOrder MeasureTheory.NullSingletonClass

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Cites13

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
PartialOrderstatement and proof · cited by 6,410
MeasureTheory.integralstatement · cited by 1,779
MeasureTheory.Measure.restrictstatement · cited by 1,646
Set.Ioostatement · cited by 1,214
Set.Iocstatement · cited by 971
MeasureTheory.NullSingletonClassstatement and proof · cited by 125
MeasureTheory.NullSingletonClass.measure_singletonproof · cited by 47

Cited by6

Results whose statement or proof uses this declaration.

intervalIntegral.integral_mono_on_of_le_Iooproof · cited by 4
MeasureTheory.integral_Ico_eq_integral_Iocproof · cited by 2
MeasureTheory.posConvolution_eq_convolution_indicatorproof · cited by 2
WeakFEPair.f_modif_aux2proof · cited by 1
ProbabilityTheory.lintegral_betaPDF_eq_oneproof · cited by 1
intervalIntegral.integrableOn_deriv_right_of_nonnegproof · cited by 1