Theorems · Theorem · measure theory
MeasureTheory.mul_meas_ge_le_integral_of_nonneg
∀ {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ},
0 ≤ᵐ[μ] f → MeasureTheory.Integrable f μ → ∀ (ε : ℝ), ε * μ.real {x | ε ≤ f x} ≤ ∫ (x : α), f x ∂μMarkov's inequality also known as Chebyshev's first inequality.
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 256 from the axioms · uses propext, Classical.choice, Quot.sound
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Cites29
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
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- Top.topproof · cited by 9,680
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- MulZeroClass.mul_zeroproof · cited by 2,091
- MeasureTheory.integralstatement and proof · cited by 1,779
- MeasureTheory.Integrablestatement and proof · cited by 1,367
Cited by1
Results whose statement or proof uses this declaration.
- ProbabilityTheory.measure_ge_le_exp_mul_mgfproof · cited by 3