Theorems · Theorem · measure theory

integral_withDensity_eq_integral_toReal_smul

∀ {X : Type u_1} {E : Type u_3} [inst : MeasurableSpace X] {μ : MeasureTheory.Measure X} [inst_1 : NormedAddCommGroup E]
  [inst_2 : NormedSpace ℝ E] {f : X → ENNReal},
  Measurable f →
    (∀ᵐ (x : X) ∂μ, f x < ⊤) → ∀ (g : X → E), ∫ (x : X), g x ∂μ.withDensity f = ∫ (x : X), (f x).toReal • g x ∂μ

Defined in: Mathlib.MeasureTheory.Integral.Bochner.ContinuousLinearMap
Cited by: 2 results in Mathlib
Foundations: Depth 264 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: MeasurableSpace NormedAddCommGroup NormedSpace

Mathlib docs Source Loogle LeanSearch

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.

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
ENNRealstatement and proof · cited by 9,879
Top.topstatement and proof · cited by 9,680
Filter.Eventuallystatement and proof · cited by 3,134
MeasureTheory.aestatement and proof · cited by 2,352
MeasureTheory.integralstatement · cited by 1,779
Measurablestatement and proof · cited by 1,499
ENNReal.toRealstatement · cited by 859

Cited by2

Results whose statement or proof uses this declaration.

MeasureTheory.integral_rnDeriv_smulproof · cited by 5
ProbabilityTheory.integral_gaussianReal_eq_integral_smulproof · cited by 1