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Explanation legend Lean4 Domain Explanation Language (L4DEL)

L4DEL explanations are a separate layer. They do not change the Lean source. Intuition is explanatory prose; numbered items are a reading path; sources link the wording back to evidence.

L4DEL: Lean4 Domain Explanation Language

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13 declarations, 0 contain sorry, 0 declared as axioms

Lorentz space

This file describes properties of almost everywhere strongly measurable functions with finite (p,q)-seminorm, denoted by eLorentzNorm f p q μ.

The Prop-valued MemLorentz f p q μ states that a function f : α → ε has finite (p,q)-seminorm and is almost everywhere strongly measurable.

Main definitions

TODO

Imports (1)

Carleson.ToMathlib.WeakType

Imported by 2 files

def MeasureTheory.eLorentzNorm' The Lorentz seminorm of a function, for 0 < p < ∞ line 34 source
Source documentation
The Lorentz seminorm of a function, for 0 < p < ∞

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: MeasurableSpace, ENorm, ENNReal, MeasureTheory.Measure
Proof uses: HMul.hMul, HPow.hPow, Real, ENNReal.toReal, Inv.inv, MeasureTheory.eLpNorm, NNReal, MeasureTheory.distribution, MeasureTheory.Measure.withDensity, MeasureTheory.MeasureSpace.volume
12 instances, coercions and auxiliary constants left out.
```
def eLorentzNorm' (f : α → ε) (p : ℝ≥0∞) (q : ℝ≥0∞) (μ : Measure α) : ℝ≥0∞ :=
…
```
lemma MeasureTheory.eLorentzNorm'_exponent_zero' line 39 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: MeasurableSpace, ENNReal, ENorm, MeasureTheory.Measure, Eq, MeasureTheory.eLorentzNorm', OfNat.ofNat
Proof uses: of_eq_true, Eq.trans, True, congrFun', congrArg, HMul.hMul, HPow.hPow, Real, ENNReal.toReal, Inv.inv, MeasureTheory.eLpNorm, NNReal, MeasureTheory.distribution, MeasureTheory.Measure.withDensity, MeasureTheory.MeasureSpace.volume, congr, Top.top, ENNReal.inv_zero, ENNReal.rpow_zero, autoParam and 5 more
24 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma MeasureTheory.eLorentzNorm'_eq_integral_distribution_rpow line 42 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: ENNReal, ENorm, MeasurableSpace, MeasureTheory.Measure, Eq, MeasureTheory.eLorentzNorm', OfNat.ofNat, HMul.hMul, MeasureTheory.lintegral, NNReal, MeasureTheory.MeasureSpace.volume, HPow.hPow, Real, MeasureTheory.distribution, Inv.inv, ENNReal.toReal
Proof uses: id, Eq.mpr, MeasureTheory.eLpNorm, MeasureTheory.Measure.withDensity, congrFun', congrArg, congr, Eq.trans, inv_one, ENNReal.rpow_one, autoParam, funext, ENNReal.toReal_inv, HMul, Eq.rec, Eq.refl, ENorm.enorm, HDiv.hDiv, MeasureTheory.eLpNorm_eq_lintegral_rpow_enorm_toReal, Mathlib.Meta.NormNum.isNat_eq_false and 52 more
116 instances, coercions and auxiliary constants left out.

Tactic trace: 28 steps
Open to load the checked before-and-after goal states.
def MeasureTheory.eLorentzNorm The Lorentz seminorm of a function line 67 source
Source documentation
The Lorentz seminorm of a function

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: MeasurableSpace, ENorm, ENNReal, MeasureTheory.Measure
Proof uses: ite, Eq, OfNat.ofNat, Top.top, MeasureTheory.eLpNormEssSup, HMul.hMul, MeasureTheory.eLorentzNorm'
10 instances, coercions and auxiliary constants left out.
```
def eLorentzNorm (f : α → ε) (p q : ℝ≥0∞) (μ : Measure α) : ℝ≥0∞ :=
…
```
lemma MeasureTheory.eLorentzNorm_eq_eLorentzNorm' line 74 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: MeasurableSpace, ENNReal, ENorm, MeasureTheory.Measure, Ne, OfNat.ofNat, Top.top, Eq, MeasureTheory.eLorentzNorm, MeasureTheory.eLorentzNorm'
Proof uses: id, of_eq_true, ite, MeasureTheory.eLpNormEssSup, HMul.hMul, Eq.trans, True, congrFun', congrArg, ite_eq_right_of_eq_false, eq_false, eq_self
13 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
lemma MeasureTheory.eLorentzNorm_exponent_zero line 80 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: MeasurableSpace, ENNReal, ENorm, MeasureTheory.Measure, Eq, MeasureTheory.eLorentzNorm, OfNat.ofNat
Proof uses: of_eq_true, Eq.trans, True, congrFun', congrArg, ite_eq_left_of_eq_true, ite, Top.top, MeasureTheory.eLpNormEssSup, HMul.hMul, MeasureTheory.eLorentzNorm', eq_self
12 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma MeasureTheory.eLorentzNorm_exponent_zero' line 83 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: MeasurableSpace, ENNReal, ENorm, MeasureTheory.Measure, Eq, MeasureTheory.eLorentzNorm, OfNat.ofNat
Proof uses: of_eq_true, Eq.trans, True, congrFun', congrArg, ite, Top.top, MeasureTheory.eLpNormEssSup, HMul.hMul, MeasureTheory.eLorentzNorm', ite_congr, Eq.refl, Not, ite.congr_simp, ite_eq_left_of_eq_true, eq_self, HPow.hPow, Real, ENNReal.toReal, Inv.inv and 14 more
26 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma MeasureTheory.eLorentzNorm_exponent_top_top line 87 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: MeasurableSpace, ENorm, MeasureTheory.Measure, Eq, ENNReal, MeasureTheory.eLorentzNorm, Top.top, MeasureTheory.eLpNormEssSup
Proof uses: of_eq_true, Eq.trans, True, congrFun', congrArg, ite, OfNat.ofNat, HMul.hMul, MeasureTheory.eLorentzNorm', ite_eq_right_of_eq_false, ite_eq_left_of_eq_true, eq_self
12 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma MeasureTheory.eLorentzNorm_exponent_top' line 90 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: MeasurableSpace, ENNReal, ENorm, MeasureTheory.Measure, Ne, OfNat.ofNat, Top.top, MeasureTheory.eLpNormEssSup, Eq, MeasureTheory.eLorentzNorm
Proof uses: Eq.mpr, ite, HMul.hMul, id, congrFun', congrArg, Eq.trans, MeasureTheory.eLorentzNorm', ite_eq_right_of_eq_false, ite_eq_left_of_eq_true, eq_self, False, Not, Iff, iff_false, ENNReal.top_mul, Eq.refl
15 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
lemma MeasureTheory.eLorentzNorm_exponent_top line 95 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: MeasurableSpace, ENNReal, MeasureTheory.Measure, TopologicalSpace, ENormedAddMonoid, Ne, OfNat.ofNat, Top.top, Not, Filter.EventuallyEq, MeasureTheory.ae, Eq, MeasureTheory.eLorentzNorm
Proof uses: MeasureTheory.eLorentzNorm_exponent_top', Mathlib.Tactic.Contrapose.contrapose₄, MeasureTheory.eLpNormEssSup, Iff.mp, MeasureTheory.eLpNormEssSup_eq_zero_iff
26 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
def MeasureTheory.MemLorentz A function is in the Lorentz space L^{p,q} if it is (strongly a.e.)-measurable and has finite Lorentz seminorm line 104 source
Source documentation
A function is in the Lorentz space L^{p,q} if it is (strongly a.e.)-measurable and has finite Lorentz seminorm.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: MeasurableSpace, ENorm, TopologicalSpace, ENNReal, MeasureTheory.Measure
Proof uses: And, MeasureTheory.AEStronglyMeasurable, LT.lt, MeasureTheory.eLorentzNorm, Top.top
4 instances, coercions and auxiliary constants left out.
```
def MemLorentz [TopologicalSpace ε] (f : α → ε) (p r : ℝ≥0∞) (μ : Measure α) : Prop :=
…
```
theorem MeasureTheory.MemLorentz.aestronglyMeasurable line 107 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: MeasurableSpace, ENNReal, ENorm, MeasureTheory.Measure, TopologicalSpace, MeasureTheory.MemLorentz, MeasureTheory.AEStronglyMeasurable
Proof uses: And.left, LT.lt, MeasureTheory.eLorentzNorm, Top.top
4 instances, coercions and auxiliary constants left out.
lemma MeasureTheory.MemLorentz.aemeasurable line 112 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: MeasurableSpace, ENNReal, ENorm, MeasureTheory.Measure, TopologicalSpace, TopologicalSpace.PseudoMetrizableSpace, BorelSpace, MeasureTheory.MemLorentz, AEMeasurable
Proof uses: MeasureTheory.AEStronglyMeasurable.aemeasurable, MeasureTheory.MemLorentz.aestronglyMeasurable