Theorems · Theorem · measure theory
MeasureTheory.volume_pi_closedBall
∀ {ι : Type u_1} {α : ι → Type u_3} [inst : Fintype ι] [inst_1 : (i : ι) → MeasureTheory.MeasureSpace (α i)]
[∀ (i : ι), MeasureTheory.SigmaFinite MeasureTheory.volume] [inst_3 : (i : ι) → MetricSpace (α i)] (x : (i : ι) → α i)
{r : ℝ}, 0 ≤ r → MeasureTheory.volume (Metric.closedBall x r) = ∏ i, MeasureTheory.volume (Metric.closedBall (x i) r)- Defined in
- Mathlib.MeasureTheory.Constructions.Pi
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 232 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites14
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coestatement · cited by 62,936
- Setstatement · cited by 53,352
- Realstatement and proof · cited by 25,697
- MeasureTheory.Measurestatement · cited by 10,939
- ENNRealstatement · cited by 9,879
- Fintypestatement and proof · cited by 7,736
- Finset.univstatement · cited by 3,473
- Finset.prodstatement · cited by 2,356
- MetricSpacestatement and proof · cited by 1,684
- MeasureTheory.MeasureSpace.volumestatement and proof · cited by 1,323
- Metric.closedBallstatement · cited by 704
- MeasureTheory.SigmaFinitestatement and proof · cited by 526
Cited by1
Results whose statement or proof uses this declaration.
- Real.volume_pi_closedBallproof · cited by 0