Theorems · Theorem · measure theory
MeasureTheory.integral_eq_integral_pos_part_sub_integral_neg_part
∀ {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ},
MeasureTheory.Integrable f μ → ∫ (a : α), f a ∂μ = ∫ (a : α), ↑(f a).toNNReal ∂μ - ∫ (a : α), ↑(-f a).toNNReal ∂μ- Cited by
- 3 results in Mathlib
- Foundations
- Depth 253 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
- MeasureTheory.integralstatement and proof · cited by 1,779
- MeasureTheory.Integrablestatement and proof · cited by 1,367
- NNReal.toRealstatement · cited by 1,260
- Real.toNNRealstatement · cited by 267
- MeasureTheory.integral_subproof · cited by 55
- MeasureTheory.Integrable.negproof · cited by 40
- max_zero_sub_max_neg_zero_eq_selfproof · cited by 6
- MeasureTheory.Integrable.real_toNNRealproof · cited by 3
Cited by3
Results whose statement or proof uses this declaration.
- MeasureTheory.Measure.ext_of_integral_eq_on_compactlySupported_nnrealproof · cited by 2
- MeasureTheory.exists_lt_lowerSemicontinuous_integral_ltproof · cited by 2
- MeasureTheory.eLpNorm_one_le_of_leproof · cited by 1