Theorems · Theorem · measure theory
MeasureTheory.exists_le_average
∀ {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ} [MeasureTheory.IsFiniteMeasure μ],
μ ≠ 0 → MeasureTheory.Integrable f μ → ∃ x, f x ≤ ⨍ (a : α), f a ∂μFirst moment method. The minimum of an integrable function is smaller than its mean.
- Defined in
- Mathlib.MeasureTheory.Integral.Average
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 259 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites11
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
- MeasurableSpacestatement and proof · cited by 13,106
- MeasureTheory.Measurestatement and proof · cited by 10,939
- Set.ofPredproof · cited by 6,101
- Set.Nonemptyproof · cited by 2,627
- LT.lt.ne'proof · cited by 1,417
- MeasureTheory.Integrablestatement and proof · cited by 1,367
- MeasureTheory.IsFiniteMeasurestatement and proof · cited by 1,078
- MeasureTheory.averagestatement and proof · cited by 87
- MeasureTheory.nonempty_of_measure_ne_zeroproof · cited by 22
- MeasureTheory.measure_le_average_posproof · cited by 3
Cited by1
Results whose statement or proof uses this declaration.
- MeasureTheory.exists_le_integralproof · cited by 0