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fix: disambiguate duplicate Verso labels in MeasureTheory 1.2.2-1.4.3 #660

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fix: disambiguate duplicate Verso labels in MeasureTheory 1.2.2-1.4.3 #660

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4 changes: 2 additions & 2 deletions Analysis/MeasureTheory/Section_1_2_2.lean
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Expand Up @@ -1744,12 +1744,12 @@ theorem Lebesgue_measure.translate {d:ℕ} {E: Set (EuclideanSpace' d)} (x: Eucl
(hE: LebesgueMeasurable E): Lebesgue_measure (E + {x}) = Lebesgue_measure E := by
sorry

/-- Exercise 1.2.21 (Change of variables) -/
/-- Exercise 1.2.21 (Change of variables, measurability) -/
lemma LebesgueMeasurable.linear {d:ℕ} (T: EuclideanSpace' d ≃ₗ[ℝ] EuclideanSpace' d)
{E: Set (EuclideanSpace' d)} (hE: LebesgueMeasurable E): LebesgueMeasurable (T '' E) := by
sorry

/-- Exercise 1.2.21 (Change of variables) -/
/-- Exercise 1.2.21 (Change of variables, the measure scales by the determinant) -/
lemma Lebesgue_measure.linear {d:ℕ} (A: Matrix (Fin d) (Fin d) ℝ) [Invertible A]
{E: Set (EuclideanSpace' d)} (hE: LebesgueMeasurable E): Lebesgue_measure (A.linear_equiv '' E) = |A.det| * Lebesgue_measure E := by
sorry
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28 changes: 14 additions & 14 deletions Analysis/MeasureTheory/Section_1_3_1.lean
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Expand Up @@ -819,15 +819,15 @@ lemma UnsignedSimpleFunction.integral_eq {d:ℕ} {f: EuclideanSpace' d → EReal
simp only [UnsignedSimpleFunction.IntegralWellDef.weightedMeasureSum] at h
exact h.symm

/-- Definition 1.3.5 -/
/-- Definition 1.3.5 (almost always) -/
def AlmostAlways {d:ℕ} (P: EuclideanSpace' d → Prop) : Prop :=
IsNull { x | ¬ P x }

/-- Definition 1.3.5 -/
/-- Definition 1.3.5 (almost everywhere equal) -/
def AlmostEverywhereEqual {d:ℕ} {X: Type*} (f g: EuclideanSpace' d → X) : Prop :=
AlmostAlways (fun x ↦ f x = g x)

/-- Definition 1.3.5 -/
/-- Definition 1.3.5 (support) -/
def Support {X Y: Type*} [Zero Y] (f: X → Y) : Set X := { x | f x ≠ 0 }

lemma UnsignedSimpleFunction.support_measurable {d:ℕ} {f: EuclideanSpace' d → EReal} (hf: UnsignedSimpleFunction f) : LebesgueMeasurable (Support f) := by
Expand Down Expand Up @@ -1027,12 +1027,12 @@ theorem AlmostEverywhereEqual.equivalence {d:ℕ} {X: Type*} :
Equivalence (@AlmostEverywhereEqual d X) :=
⟨refl, symm, trans⟩

/-- Exercise 1.3.1 (i) (Unsigned linearity) -/
/-- Exercise 1.3.1 (i) (Unsigned linearity, sum) -/
lemma UnsignedSimpleFunction.integral_add {d:ℕ} {f g: EuclideanSpace' d → EReal} (hf: UnsignedSimpleFunction f) (hg: UnsignedSimpleFunction g) :
(hf.add hg).integ = hf.integ + hg.integ := by
sorry

/-- Exercise 1.3.1 (i) (Unsigned linearity) -/
/-- Exercise 1.3.1 (i) (Unsigned linearity, scalar multiple) -/
lemma UnsignedSimpleFunction.integral_smul {d:ℕ} {f: EuclideanSpace' d → EReal} (hf: UnsignedSimpleFunction f) {c:EReal} (hc: c ≥ 0) :
(hf.smul hc).integ = c * hf.integ := by
sorry
Expand All @@ -1059,12 +1059,12 @@ lemma UnsignedSimpleFunction.integral_le_integral_of_aeLe {d:ℕ} {f g: Euclidea
hf.integ ≤ hg.integ := by
sorry

/-- Exercise 1.3.1(vi) (Compatibility with Lebesgue measure) -/
/-- Exercise 1.3.1(vi) (Compatibility with Lebesgue measure, indicator) -/
lemma UnsignedSimpleFunction.indicator {d:ℕ} {E: Set (EuclideanSpace' d)} (hE: LebesgueMeasurable E) :
UnsignedSimpleFunction (Real.toEReal ∘ E.indicator') := by
sorry

/-- Exercise 1.3.1(vi) (Compatibility with Lebesgue measure) -/
/-- Exercise 1.3.1(vi) (Compatibility with Lebesgue measure, integral of an indicator) -/
lemma UnsignedSimpleFunction.integral_indicator {d:ℕ} {E: Set (EuclideanSpace' d)} (hE: LebesgueMeasurable E) :
(UnsignedSimpleFunction.indicator hE).integ = Lebesgue_measure E := by
sorry
Expand All @@ -1075,11 +1075,11 @@ lemma RealSimpleFunction.abs {d:ℕ} {f: EuclideanSpace' d → ℝ} (hf: RealSim
lemma ComplexSimpleFunction.abs {d:ℕ} {f: EuclideanSpace' d → ℂ} (hf: ComplexSimpleFunction f) : UnsignedSimpleFunction (EReal.abs_fun f) := by
sorry

/-- Definition 1.3.6 (Absolutely convergent simple integral) -/
/-- Definition 1.3.6 (Absolutely convergent simple integral, real) -/
def RealSimpleFunction.AbsolutelyIntegrable {d:ℕ} {f: EuclideanSpace' d → ℝ} (hf: RealSimpleFunction f) : Prop :=
(hf.abs).integ < ⊤

/-- Definition 1.3.6 (Absolutely convergent simple integral) -/
/-- Definition 1.3.6 (Absolutely convergent simple integral, complex) -/
def ComplexSimpleFunction.AbsolutelyIntegrable {d:ℕ} {f: EuclideanSpace' d → ℂ} (hf: ComplexSimpleFunction f) : Prop :=
(hf.abs).integ < ⊤

Expand Down Expand Up @@ -1272,20 +1272,20 @@ lemma ComplexSimpleFunction.AbsolutelyIntegrable.smul {d:ℕ} {f: EuclideanSpace
lemma ComplexSimpleFunction.AbsolutelyIntegrable.conj {d:ℕ} {f: EuclideanSpace' d → ℂ} {hf: ComplexSimpleFunction f} (hf_integ: hf.AbsolutelyIntegrable) :
(hf.conj).AbsolutelyIntegrable := by sorry

/-- Exercise 1.3.2 (i) ({lit}`*`-linearity) -/
/-- Exercise 1.3.2 (i) ({lit}`*`-linearity, sum) -/
lemma RealSimpleFunction.integ_add {d:ℕ} {f g: EuclideanSpace' d → ℝ} {hf: RealSimpleFunction f} {hg: RealSimpleFunction g} (hf_integ: hf.AbsolutelyIntegrable) (hg_integ: hg.AbsolutelyIntegrable) : (hf.add hg).integ = hf.integ + hg.integ := by sorry

lemma ComplexSimpleFunction.integ_add {d:ℕ} {f g: EuclideanSpace' d → ℂ} {hf: ComplexSimpleFunction f} {hg: ComplexSimpleFunction g} (hf_integ: hf.AbsolutelyIntegrable) (hg_integ: hg.AbsolutelyIntegrable) : (hf.add hg).integ = hf.integ + hg.integ := by
sorry

/-- Exercise 1.3.2 (i) ({lit}`*`-linearity) -/
/-- Exercise 1.3.2 (i) ({lit}`*`-linearity, scalar multiple) -/
lemma RealSimpleFunction.integ_smul {d:ℕ} {f: EuclideanSpace' d → ℝ} {hf: RealSimpleFunction f} (hf_integ: hf.AbsolutelyIntegrable) (a: ℝ) : (hf.smul a).integ = a * hf.integ := by
sorry

lemma ComplexSimpleFunction.integ_smul {d:ℕ} {f: EuclideanSpace' d → ℂ} {hf: ComplexSimpleFunction f} (hf_integ: hf.AbsolutelyIntegrable) (a: ℂ) : (hf.smul a).integ = a * hf.integ := by
sorry

/-- Exercise 1.3.2 (i) ({lit}`*`-linearity) -/
/-- Exercise 1.3.2 (i) ({lit}`*`-linearity, conjugation) -/
lemma ComplexSimpleFunction.integral_conj {d:ℕ} {f: EuclideanSpace' d → ℂ} {hf: ComplexSimpleFunction f} (hf_integ: hf.AbsolutelyIntegrable) : (hf.conj).integ = (starRingEnd ℂ) hf.integ := by
sorry

Expand All @@ -1296,7 +1296,7 @@ lemma RealSimpleFunction.integral_eq_integral_of_aeEqual {d:ℕ} {f g: Euclidean
lemma ComplexSimpleFunction.integral_eq_integral_of_aeEqual {d:ℕ} {f g: EuclideanSpace' d → ℂ} {hf: ComplexSimpleFunction f} {hg: ComplexSimpleFunction g} (hf_integ: hf.AbsolutelyIntegrable) (hg_integ: hg.AbsolutelyIntegrable) (h_ae: AlmostEverywhereEqual f g) : hf.integ = hg.integ := by
sorry

/-- Exercise 1.3.2(iii) (Compatibility with Lebesgue measure) -/
/-- Exercise 1.3.2(iii) (Compatibility with Lebesgue measure, indicator) -/
lemma RealSimpleFunction.indicator {d:ℕ} {E: Set (EuclideanSpace' d)} (hE: LebesgueMeasurable E) :
RealSimpleFunction (E.indicator') := by
sorry
Expand All @@ -1305,7 +1305,7 @@ lemma ComplexSimpleFunction.indicator {d:ℕ} {E: Set (EuclideanSpace' d)} (hE:
ComplexSimpleFunction (Complex.indicator E) := by
sorry

/-- Exercise 1.3.2(iii) (Compatibility with Lebesgue measure) -/
/-- Exercise 1.3.2(iii) (Compatibility with Lebesgue measure, integral of an indicator) -/
lemma RealSimpleFunction.integral_indicator {d:ℕ} {E: Set (EuclideanSpace' d)} (hE: LebesgueMeasurable E) (hfin: Lebesgue_measure E < ⊤): (RealSimpleFunction.indicator hE).integ = (Lebesgue_measure E).toReal := by
sorry

Expand Down
4 changes: 2 additions & 2 deletions Analysis/MeasureTheory/Section_1_4_1.lean
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Expand Up @@ -29,7 +29,7 @@ instance ConcreteBooleanAlgebra.instPartialOrder (X:Type*) : PartialOrder (Concr
def ConcreteBooleanAlgebra.measurableSets {X:Type*} (B: ConcreteBooleanAlgebra X) : Set (Set X) :=
{ E | B.measurable E }

/-- Example 1.4.3 -/
/-- Example 1.4.3 (the largest algebra) -/
instance ConcreteBooleanAlgebra.instOrderTop {X:Type*} : OrderTop (ConcreteBooleanAlgebra X) :=
{
top := {
Expand All @@ -41,7 +41,7 @@ instance ConcreteBooleanAlgebra.instOrderTop {X:Type*} : OrderTop (ConcreteBoole
le_top := sorry
}

/-- Example 1.4.3 -/
/-- Example 1.4.3 (the smallest algebra) -/
instance ConcreteBooleanAlgebra.instOrderBot {X:Type*} : OrderBot (ConcreteBooleanAlgebra X) :=
{
bot := {
Expand Down
10 changes: 5 additions & 5 deletions Analysis/MeasureTheory/Section_1_4_3.lean
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Expand Up @@ -17,7 +17,7 @@ class FinitelyAdditiveMeasure {X:Type*} (B: ConcreteBooleanAlgebra X) where
measure_finite_additive : ∀ E F : Set X, B.measurable E → B.measurable F → Disjoint E F →
measure (E ∪ F) = measure E + measure F

/-- Example 1.4.21 -/
/-- Example 1.4.21 (Lebesgue measure) -/
noncomputable def FinitelyAdditiveMeasure.lebesgue (d:ℕ) : FinitelyAdditiveMeasure (LebesgueMeasurable.boolean_algebra d) :=
{
measure A := Lebesgue_measure A
Expand All @@ -26,7 +26,7 @@ noncomputable def FinitelyAdditiveMeasure.lebesgue (d:ℕ) : FinitelyAdditiveMea
measure_finite_additive := by sorry
}

/-- Example 1.4.21 -/
/-- Example 1.4.21 (restriction to a subalgebra) -/
def FinitelyAdditiveMeasure.restrict_alg {X:Type*} {B: ConcreteBooleanAlgebra X} (μ: FinitelyAdditiveMeasure B) {B':ConcreteBooleanAlgebra X} (hBB': B' ≤ B) : FinitelyAdditiveMeasure B' :=
{
measure := μ.measure
Expand All @@ -35,15 +35,15 @@ def FinitelyAdditiveMeasure.restrict_alg {X:Type*} {B: ConcreteBooleanAlgebra X}
measure_finite_additive := by sorry
}

/-- Example 1.4.21 -/
/-- Example 1.4.21 (Jordan measure) -/
noncomputable def FinitelyAdditiveMeasure.jordan (d:ℕ) : FinitelyAdditiveMeasure (JordanMeasurable.boolean_algebra d) :=
(FinitelyAdditiveMeasure.lebesgue d).restrict_alg (LebesgueMeasurable.gt_jordan_boolean_algebra d)

/-- Example 1.4.21 -/
/-- Example 1.4.21 (null sets) -/
noncomputable def FinitelyAdditiveMeasure.null (d:ℕ) : FinitelyAdditiveMeasure (IsNull.boolean_algebra d) :=
(FinitelyAdditiveMeasure.lebesgue d).restrict_alg (IsNull.lt_lebesgue_boolean_algebra d)

/-- Example 1.4.21 -/
/-- Example 1.4.21 (elementary sets) -/
noncomputable def FinitelyAdditiveMeasure.elem (d:ℕ) : FinitelyAdditiveMeasure (EuclideanSpace'.elementary_boolean_algebra d) :=
(FinitelyAdditiveMeasure.lebesgue d).restrict_alg (by sorry)

Expand Down
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