Theorems · Theorem · measure theory

MeasureTheory.setIntegral_neg_eq_setIntegral_nonpos

∀ {X : Type u_1} {E : Type u_3} {mX : MeasurableSpace X} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E]
  {μ : MeasureTheory.Measure X} [inst_2 : PartialOrder E] {f : X → E},
  MeasureTheory.AEStronglyMeasurable f μ → ∫ (x : X) in {x | f x < 0}, f x ∂μ = ∫ (x : X) in {x | f x ≤ 0}, f x ∂μ

Defined in: Mathlib.MeasureTheory.Integral.Bochner.Set
Cited by: 1 results in Mathlib
Foundations: Depth 258 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: NormedAddCommGroup NormedSpace PartialOrder

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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
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
Set.ofPredstatement and proof · cited by 6,101
MeasureTheory.integralstatement and proof · cited by 1,779
Filter.univ_mem'proof · cited by 1,672
MeasureTheory.Measure.restrictstatement and proof · cited by 1,646
Filter.mp_memproof · cited by 1,537

Cited by1

Results whose statement or proof uses this declaration.

MeasureTheory.integral_norm_eq_pos_sub_negproof · cited by 1