Theorems · Theorem · real analysis
AntitoneOn.sum_range_le_integral
∀ {f : ℝ → ℝ} {N : ℕ},
AntitoneOn f (Set.Icc 0 ↑N) →
MeasureTheory.IntegrableOn f (Set.Ioi 0) MeasureTheory.volume →
(∀ t ∈ Set.Ioi 0, 0 ≤ f t) → ∑ n ∈ Finset.range N, f ↑(n + 1) ≤ ∫ (x : ℝ) in Set.Ioi 0, f xThe partial sums of a nonnegative function are bounded by the integral over (0, ∞).
- Defined in
- Mathlib.Analysis.SumIntegralComparisons
- Cited by
- 0 results in Mathlib
- Foundations
- Depth 262 from the axioms · uses propext, Classical.choice, Quot.sound
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Cites16
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Setstatement · cited by 53,352
- Realstatement and proof · cited by 25,697
- Finsetproof · cited by 13,712
- Finset.sumstatement and proof · cited by 5,195
- Nat.cast_zeroproof · cited by 1,870
- MeasureTheory.integralstatement and proof · cited by 1,779
- Set.Iccstatement and proof · cited by 1,702
- MeasureTheory.Measure.restrictstatement and proof · cited by 1,646
- Set.Ioistatement and proof · cited by 1,463
- Finset.rangestatement · cited by 1,341
- MeasureTheory.MeasureSpace.volumestatement and proof · cited by 1,323
- MeasureTheory.IntegrableOnstatement and proof · cited by 548
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