Theorems · Theorem · real analysis
AntitoneOn.integrableOn_Ioi_of_summable_comp_add
∀ {f : ℝ → ℝ} {N : ℕ},
AntitoneOn f (Set.Ici ↑N) →
(Summable fun n => f ↑(n + N)) →
(∀ t ∈ Set.Ioi ↑N, 0 ≤ f t) → MeasureTheory.IntegrableOn f (Set.Ioi ↑N) MeasureTheory.volumeConverse to the integral test: a nonnegative, integrable, summable function is integrable.
- Defined in
- Mathlib.Analysis.SumIntegralComparisons
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 260 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites29
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
- NormedAddCommGroupproof · cited by 15,752
- NormedSpaceproof · cited by 12,499
- MeasureTheory.Measureproof · cited by 10,939
- Finset.sumproof · cited by 5,195
- Filter.atTopproof · cited by 2,405
- Finset.sum_congrproof · cited by 2,323
- SummationFilter.unconditionalstatement and proof · cited by 2,068
- Filter.univ_mem'proof · cited by 1,672
- Filter.mp_memproof · cited by 1,537
- Set.Ioistatement and proof · cited by 1,463
Cited by2
Results whose statement or proof uses this declaration.
- AntitoneOn.integral_le_tsum_comp_addproof · cited by 1
- AntitoneOn.integrableOn_Ioi_zero_of_summableproof · cited by 0