Theorems · Theorem · measure theory

MeasureTheory.ae_norm_ofReal_f_le_bound

∀ {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [inst : NormedAddCommGroup β]
  {F : ℕ → α → β} {f : α → β} {bound : α → ℝ},
  (∀ (n : ℕ), ∀ᵐ (a : α) ∂μ, ‖F n a‖ ≤ bound a) →
    (∀ᵐ (a : α) ∂μ, Filter.Tendsto (fun n => F n a) Filter.atTop (nhds (f a))) →
      ∀ᵐ (a : α) ∂μ, ENNReal.ofReal ‖f a‖ ≤ ENNReal.ofReal (bound a)

Defined in: Mathlib.MeasureTheory.Function.L1Space.HasFiniteIntegral
Cited by: 3 results in Mathlib
Foundations: Depth 173 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: NormedAddCommGroup

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Cites18

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

Realstatement and proof · cited by 25,697
NormedAddCommGroupstatement and proof · cited by 15,752
MeasurableSpacestatement and proof · cited by 13,106
MeasureTheory.Measurestatement and proof · cited by 10,939
ENNRealstatement · cited by 9,879
nhdsstatement and proof · cited by 5,554
Norm.normstatement and proof · cited by 5,413
Filter.Tendstostatement and proof · cited by 3,814
Filter.Eventuallystatement and proof · cited by 3,134
Filter.atTopstatement and proof · cited by 2,405
MeasureTheory.aestatement and proof · cited by 2,352
ENNReal.ofRealstatement and proof · cited by 863

Cited by3

Results whose statement or proof uses this declaration.

MeasureTheory.tendsto_lintegral_norm_of_dominated_convergenceproof · cited by 2
MeasureTheory.hasFiniteIntegral_of_dominated_convergenceproof · cited by 1
MeasureTheory.all_ae_ofReal_f_le_boundproof · cited by 0