Theorems · Theorem · measure theory

MeasureTheory.SimpleFunc.exists_lt_lintegral_simpleFunc_of_lt_lintegral

∀ {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [MeasureTheory.SigmaFinite μ]
  {f : MeasureTheory.SimpleFunc α NNReal} {L : ENNReal},
  L < ∫⁻ (x : α), ↑(f x) ∂μ → ∃ g, (∀ (x : α), g x ≤ f x) ∧ ∫⁻ (x : α), ↑(g x) ∂μ < ⊤ ∧ L < ∫⁻ (x : α), ↑(g x) ∂μ

Defined in: Mathlib.MeasureTheory.Integral.Lebesgue.Countable
Cited by: 1 results in Mathlib
Foundations: Depth 204 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: MeasureTheory.SigmaFinite

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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
Setproof · cited by 53,352
MeasurableSpacestatement and proof · cited by 13,106
MeasureTheory.Measurestatement and proof · cited by 10,939
ENNRealstatement and proof · cited by 9,879
Top.topstatement and proof · cited by 9,680
NNRealstatement and proof · cited by 4,310
LE.le.transproof · cited by 3,151
MeasurableSetproof · cited by 3,075
add_zeroproof · cited by 2,707
zero_addproof · cited by 2,366
mul_commproof · cited by 2,262

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MeasureTheory.exists_lt_lintegral_simpleFunc_of_lt_lintegralproof · cited by 1