Theorems · Theorem · measure theory
MeasureTheory.SignedMeasure.exists_compl_positive_negative
∀ {α : Type u_1} [inst : MeasurableSpace α] (s : MeasureTheory.SignedMeasure α),
∃ i,
MeasurableSet i ∧
MeasureTheory.VectorMeasure.restrict 0 i ≤ MeasureTheory.VectorMeasure.restrict s i ∧
MeasureTheory.VectorMeasure.restrict s iᶜ ≤ MeasureTheory.VectorMeasure.restrict 0 iᶜAlternative formulation of MeasureTheory.SignedMeasure.exists_isCompl_positive_negative
(the Hahn decomposition theorem) using set complements.
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 155 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- MeasurableSpace
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites48
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coeproof · cited by 62,936
- Setstatement and proof · cited by 53,352
- Realstatement and proof · cited by 25,697
- MeasurableSpacestatement and proof · cited by 13,106
- Set.ofPredproof · cited by 6,101
- nhdsproof · cited by 5,554
- Filter.Tendstoproof · cited by 3,814
- MeasurableSetstatement and proof · cited by 3,075
- Compl.complstatement and proof · cited by 2,925
- Set.iUnionproof · cited by 2,483
- Filter.atTopproof · cited by 2,405
- le_antisymmproof · cited by 2,068
Cited by3
Results whose statement or proof uses this declaration.
- MeasureTheory.SignedMeasure.toJordanDecompositionproof · cited by 34
- MeasureTheory.SignedMeasure.toJordanDecomposition_specproof · cited by 5
- MeasureTheory.SignedMeasure.exists_isCompl_positive_negativeproof · cited by 0