Theorems · Theorem · probability

ProbabilityTheory.sum_prob_mem_Ioc_le

∀ {Ω : Type u_1} [inst : MeasureTheory.MeasureSpace Ω] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume]
  {X : Ω → ℝ},
  MeasureTheory.Integrable X MeasureTheory.volume →
    0 ≤ X →
      ∀ {K N : ℕ},
        K ≤ N →
          ∑ j ∈ Finset.range K, MeasureTheory.volume {ω | X ω ∈ Set.Ioc ↑j ↑N} ≤ ENNReal.ofReal ((∫ (a : Ω), X a) + 1)

Defined in: Mathlib.Probability.StrongLaw
Cited by: 1 results in Mathlib
Foundations: Depth 259 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: MeasureTheory.MeasureSpace MeasureTheory.IsProbabilityMeasure

Mathlib docs Source Loogle LeanSearch

Around this declaration

Dashed lines are statement dependencies; solid lines are citations in proofs.

Cites77

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 · cited by 53,352
Realstatement and proof · cited by 25,697
NormedAddCommGroupproof · cited by 15,752
NormedSpaceproof · cited by 12,499
MeasureTheory.Measurestatement and proof · cited by 10,939
ENNRealstatement and proof · cited by 9,879
Set.ofPredstatement and proof · cited by 6,101
Finset.sumstatement and proof · cited by 5,195
mul_oneproof · cited by 3,885
LE.le.transproof · cited by 3,151
Nat.cast_oneproof · cited by 2,501

Cited by1

Results whose statement or proof uses this declaration.

ProbabilityTheory.tsum_prob_mem_Ioi_lt_topproof · cited by 1