Theorems · Theorem · measure theory

MeasureTheory.setIntegral_le_nonneg

∀ {X : Type u_1} {mX : MeasurableSpace X} {μ : MeasureTheory.Measure X} {f : X → ℝ} {s : Set X},
  MeasurableSet s →
    MeasureTheory.StronglyMeasurable f →
      MeasureTheory.Integrable f μ → ∫ (x : X) in s, f x ∂μ ≤ ∫ (x : X) in {y | 0 ≤ f y}, f x ∂μ

Defined in: Mathlib.MeasureTheory.Integral.Bochner.Set
Cited by: 1 results in Mathlib
Foundations: Depth 257 from the axioms · uses propext, Classical.choice, Quot.sound

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Cites17

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

Setstatement and proof · cited by 53,352
Realstatement and proof · cited by 25,697
MeasurableSpacestatement and proof · cited by 13,106
MeasureTheory.Measurestatement and proof · cited by 10,939
Set.ofPredstatement and proof · cited by 6,101
MeasurableSetstatement and proof · cited by 3,075
MeasureTheory.integralstatement and proof · cited by 1,779
MeasureTheory.Measure.restrictstatement and proof · cited by 1,646
MeasureTheory.Integrablestatement and proof · cited by 1,367
Set.indicatorproof · cited by 723
MeasureTheory.StronglyMeasurablestatement and proof · cited by 363
MeasureTheory.stronglyMeasurable_constproof · cited by 47

Cited by1

Results whose statement or proof uses this declaration.

MeasureTheory.integral_norm_le_of_forall_fin_meas_integral_eqproof · cited by 1