Theorems · Theorem · measure theory

MeasureTheory.SignedMeasure.subset_positive_null_set

∀ {α : Type u_1} [inst : MeasurableSpace α] {s : MeasureTheory.SignedMeasure α} {u v w : Set α},
  MeasurableSet u →
    MeasurableSet v →
      MeasurableSet w →
        MeasureTheory.VectorMeasure.restrict 0 u ≤ MeasureTheory.VectorMeasure.restrict s u →
          s w = 0 → w ⊆ u → v ⊆ w → s v = 0

A subset v of a null-set w has zero measure if w is a subset of a positive set u.

Defined in: Mathlib.MeasureTheory.VectorMeasure.Decomposition.Jordan
Cited by: 2 results in Mathlib
Foundations: Depth 153 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: MeasurableSpace

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Cites19

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
Setstatement and proof · cited by 53,352
Realstatement and proof · cited by 25,697
MeasurableSpacestatement and proof · cited by 13,106
LE.le.transproof · cited by 3,151
MeasurableSetstatement and proof · cited by 3,075
Nat.cast_zeroproof · cited by 1,870
MeasureTheory.VectorMeasurestatement · cited by 451
Set.sdiff_subsetproof · cited by 156
sub_eq_zero_of_eqproof · cited by 154
MeasureTheory.VectorMeasure.restrictstatement and proof · cited by 139
MeasureTheory.SignedMeasurestatement and proof · cited by 108

Cited by2

Results whose statement or proof uses this declaration.

MeasureTheory.SignedMeasure.of_inter_eq_of_symmDiff_eq_zero_positiveproof · cited by 2
MeasureTheory.SignedMeasure.subset_negative_null_setproof · cited by 0