Theorems · Theorem · measure theory

BoundedContinuousFunction.toReal_lintegral_coe_eq_integral

∀ {X : Type u_1} [inst : TopologicalSpace X] [inst_1 : MeasurableSpace X] [OpensMeasurableSpace X]
  (f : BoundedContinuousFunction X NNReal) (μ : MeasureTheory.Measure X),
  (∫⁻ (x : X), ↑(f x) ∂μ).toReal = ∫ (x : X), ↑(f x) ∂μ

Defined in: Mathlib.MeasureTheory.Integral.BoundedContinuousFunction
Cited by: 3 results in Mathlib
Foundations: Depth 254 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: TopologicalSpace MeasurableSpace OpensMeasurableSpace

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Cites21

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

DFunLike.coestatement and proof · cited by 62,936
Realstatement and proof · cited by 25,697
TopologicalSpacestatement and proof · cited by 24,529
MeasurableSpacestatement and proof · cited by 13,106
MeasureTheory.Measurestatement and proof · cited by 10,939
NNRealstatement and proof · cited by 4,310
MeasureTheory.integralstatement · cited by 1,779
ENNReal.ofNNRealstatement and proof · cited by 1,279
NNReal.toRealstatement · cited by 1,260
MeasureTheory.lintegralstatement and proof · cited by 1,152
ENNReal.toRealstatement and proof · cited by 859
OpensMeasurableSpacestatement and proof · cited by 636

Cited by3

Results whose statement or proof uses this declaration.

MeasureTheory.FiniteMeasure.tendsto_iff_forall_integral_tendstoproof · cited by 3
MeasureTheory.ext_of_forall_integral_eq_of_IsFiniteMeasureproof · cited by 1
MeasureTheory.FiniteMeasure.ext_of_forall_integral_eqproof · cited by 0