Documentation

Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable

Functions a.e. measurable with respect to a sub-σ-algebra #

A function f verifies AEStronglyMeasurable[m] f μ if it is μ-a.e. equal to an m-strongly measurable function. This is similar to AEStronglyMeasurable, but the MeasurableSpace structures used for the measurability statement and for the measure are different.

We define lpMeas F 𝕜 m p μ, the subspace of Lp F p μ containing functions f verifying AEStronglyMeasurable[m] f μ, i.e. functions which are μ-a.e. equal to an m-strongly measurable function.

Main statements #

We define an IsometryEquiv between lpMeasSubgroup and the Lp space corresponding to the measure μ.trim hm. As a consequence, the completeness of Lp implies completeness of lpMeas.

Lp.induction_stronglyMeasurable (see also MemLp.induction_stronglyMeasurable): To prove something for an Lp function a.e. strongly measurable with respect to a sub-σ-algebra m in a normed space, it suffices to show that

@[deprecated MeasureTheory.AEStronglyMeasurable (since := "2025-01-23")]
def MeasureTheory.AEStronglyMeasurable' {α : Type u_1} {β : Type u_2} [TopologicalSpace β] (m : MeasurableSpace α) {x✝ : MeasurableSpace α} (f : αβ) (μ : Measure α) :

A function f verifies AEStronglyMeasurable[m] f μ if it is μ-a.e. equal to an m-strongly measurable function. This is similar to AEStronglyMeasurable, but the MeasurableSpace structures used for the measurability statement and for the measure are different.

Equations
@[deprecated MeasureTheory.AEStronglyMeasurable.congr (since := "2025-01-23")]
theorem MeasureTheory.AEStronglyMeasurable'.congr {α : Type u_1} {β : Type u_2} {m m0 : MeasurableSpace α} {μ : Measure α} [TopologicalSpace β] {f g : αβ} (hf : AEStronglyMeasurable f μ) (hfg : f =ᶠ[ae μ] g) :
@[deprecated MeasureTheory.AEStronglyMeasurable.mono (since := "2025-01-23")]
theorem MeasureTheory.AEStronglyMeasurable'.mono {α : Type u_1} {β : Type u_2} {m m0 : MeasurableSpace α} {μ : Measure α} [TopologicalSpace β] {f : αβ} {m' : MeasurableSpace α} (hf : AEStronglyMeasurable f μ) (hm : m m') :
@[deprecated MeasureTheory.AEStronglyMeasurable.add (since := "2025-01-23")]
theorem MeasureTheory.AEStronglyMeasurable'.add {α : Type u_1} {β : Type u_2} {m m0 : MeasurableSpace α} {μ : Measure α} [TopologicalSpace β] {f g : αβ} [Add β] [ContinuousAdd β] (hf : AEStronglyMeasurable f μ) (hg : AEStronglyMeasurable g μ) :
@[deprecated MeasureTheory.AEStronglyMeasurable.neg (since := "2025-01-23")]
theorem MeasureTheory.AEStronglyMeasurable'.neg {α : Type u_1} {β : Type u_2} {m m0 : MeasurableSpace α} {μ : Measure α} [TopologicalSpace β] [Neg β] [ContinuousNeg β] {f : αβ} (hfm : AEStronglyMeasurable f μ) :
@[deprecated MeasureTheory.AEStronglyMeasurable.sub (since := "2025-01-23")]
theorem MeasureTheory.AEStronglyMeasurable'.sub {α : Type u_1} {β : Type u_2} {m m0 : MeasurableSpace α} {μ : Measure α} [TopologicalSpace β] [AddGroup β] [IsTopologicalAddGroup β] {f g : αβ} (hfm : AEStronglyMeasurable f μ) (hgm : AEStronglyMeasurable g μ) :
@[deprecated MeasureTheory.AEStronglyMeasurable.const_smul (since := "2025-01-23")]
theorem MeasureTheory.AEStronglyMeasurable'.const_smul {α : Type u_1} {β : Type u_2} {𝕜 : Type u_3} {m m0 : MeasurableSpace α} {μ : Measure α} [TopologicalSpace β] {f : αβ} [SMul 𝕜 β] [ContinuousConstSMul 𝕜 β] (c : 𝕜) (hf : AEStronglyMeasurable f μ) :
@[deprecated MeasureTheory.AEStronglyMeasurable.const_inner (since := "2025-01-23")]
theorem MeasureTheory.AEStronglyMeasurable'.const_inner {α : Type u_1} {m m0 : MeasurableSpace α} {μ : Measure α} {𝕜 : Type u_4} {β : Type u_5} [RCLike 𝕜] [NormedAddCommGroup β] [InnerProductSpace 𝕜 β] {f : αβ} (hfm : AEStronglyMeasurable f μ) (c : β) :
AEStronglyMeasurable (fun (x : α) => inner 𝕜 c (f x)) μ
@[deprecated MeasureTheory.AEStronglyMeasurable.of_subsingleton_cod (since := "2025-01-23")]
theorem MeasureTheory.AEStronglyMeasurable'.of_subsingleton {α : Type u_1} {β : Type u_2} {m m0 : MeasurableSpace α} {μ : Measure α} [TopologicalSpace β] {f : αβ} [Subsingleton β] :
@[deprecated MeasureTheory.AEStronglyMeasurable.of_subsingleton_dom (since := "2025-01-23")]
theorem MeasureTheory.AEStronglyMeasurable'.of_subsingleton' {α : Type u_1} {β : Type u_2} {m m0 : MeasurableSpace α} {μ : Measure α} [TopologicalSpace β] {f : αβ} [Subsingleton α] :
@[deprecated MeasureTheory.AEStronglyMeasurable.mk (since := "2025-01-23")]
noncomputable def MeasureTheory.AEStronglyMeasurable'.mk {α : Type u_1} {β : Type u_2} {m m0 : MeasurableSpace α} {μ : Measure α} [TopologicalSpace β] (f : αβ) (hfm : AEStronglyMeasurable f μ) :
αβ

An m-strongly measurable function almost everywhere equal to f.

Equations
@[deprecated MeasureTheory.AEStronglyMeasurable.stronglyMeasurable_mk (since := "2025-01-23")]
@[deprecated MeasureTheory.AEStronglyMeasurable.ae_eq_mk (since := "2025-01-23")]
theorem MeasureTheory.AEStronglyMeasurable'.ae_eq_mk {α : Type u_1} {β : Type u_2} {m m0 : MeasurableSpace α} {μ : Measure α} [TopologicalSpace β] {f : αβ} (hfm : AEStronglyMeasurable f μ) :
@[deprecated Continuous.comp_aestronglyMeasurable (since := "2025-01-23")]
theorem MeasureTheory.AEStronglyMeasurable'.continuous_comp {α : Type u_1} {β : Type u_2} {m m0 : MeasurableSpace α} {μ : Measure α} [TopologicalSpace β] {γ : Type u_4} [TopologicalSpace γ] {f : αβ} {g : βγ} (hg : Continuous g) (hf : AEStronglyMeasurable f μ) :
@[deprecated MeasureTheory.AEStronglyMeasurable.of_trim (since := "2025-01-23")]
theorem MeasureTheory.aeStronglyMeasurable'_of_aeStronglyMeasurable'_trim {α : Type u_1} {β : Type u_2} {m m0 m0' : MeasurableSpace α} [TopologicalSpace β] (hm0 : m0 m0') {μ : Measure α} {f : αβ} (hf : AEStronglyMeasurable f (μ.trim hm0)) :
@[deprecated MeasureTheory.StronglyMeasurable.aestronglyMeasurable (since := "2025-01-23")]
theorem MeasureTheory.StronglyMeasurable.aeStronglyMeasurable' {α : Type u_1} {β : Type u_2} {m x✝ : MeasurableSpace α} [TopologicalSpace β] {μ : Measure α} {f : αβ} (hf : StronglyMeasurable f) :
@[deprecated MeasureTheory.ae_eq_trim_iff_of_aestronglyMeasurable (since := "2025-04-09")]

Alias of MeasureTheory.ae_eq_trim_iff_of_aestronglyMeasurable.

theorem MeasureTheory.AEStronglyMeasurable.comp_ae_measurable' {α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace β] { : MeasurableSpace α} {x✝ : MeasurableSpace γ} {f : αβ} {μ : Measure γ} {g : γα} (hf : AEStronglyMeasurable f (Measure.map g μ)) (hg : AEMeasurable g μ) :
@[deprecated MeasureTheory.AEStronglyMeasurable.of_measurableSpace_le_on (since := "2025-01-23")]
theorem MeasureTheory.AEStronglyMeasurable'.aeStronglyMeasurable'_of_measurableSpace_le_on {α : Type u_1} {E : Type u_2} {m m₂ m0 : MeasurableSpace α} {μ : Measure α} [TopologicalSpace E] [Zero E] (hm : m m0) {s : Set α} {f : αE} (hs_m : MeasurableSet s) (hs : ∀ (t : Set α), MeasurableSet (s t)MeasurableSet (s t)) (hf : AEStronglyMeasurable f μ) (hf_zero : f =ᶠ[ae (μ.restrict s)] 0) :

If the restriction to a set s of a σ-algebra m is included in the restriction to s of another σ-algebra m₂ (hypothesis hs), the set s is m measurable and a function f almost everywhere supported on s is m-ae-strongly-measurable, then f is also m₂-ae-strongly-measurable.

The subset lpMeas of Lp functions a.e. measurable with respect to a sub-sigma-algebra #

def MeasureTheory.lpMeasSubgroup {α : Type u_1} (F : Type u_2) [NormedAddCommGroup F] (m : MeasurableSpace α) [MeasurableSpace α] (p : ENNReal) (μ : Measure α) :
AddSubgroup (Lp F p μ)

lpMeasSubgroup F m p μ is the subspace of Lp F p μ containing functions f verifying AEStronglyMeasurable[m] f μ, i.e. functions which are μ-a.e. equal to an m-strongly measurable function.

Equations
def MeasureTheory.lpMeas {α : Type u_1} (F : Type u_2) (𝕜 : Type u_3) [RCLike 𝕜] [NormedAddCommGroup F] [NormedSpace 𝕜 F] (m : MeasurableSpace α) [MeasurableSpace α] (p : ENNReal) (μ : Measure α) :
Submodule 𝕜 (Lp F p μ)

lpMeas F 𝕜 m p μ is the subspace of Lp F p μ containing functions f verifying AEStronglyMeasurable[m] f μ, i.e. functions which are μ-a.e. equal to an m-strongly measurable function.

Equations
theorem MeasureTheory.mem_lpMeasSubgroup_iff_aestronglyMeasurable {α : Type u_1} {F : Type u_2} {p : ENNReal} [NormedAddCommGroup F] {m m0 : MeasurableSpace α} {μ : Measure α} {f : (Lp F p μ)} :
f lpMeasSubgroup F m p μ AEStronglyMeasurable (↑f) μ
@[deprecated MeasureTheory.mem_lpMeasSubgroup_iff_aestronglyMeasurable (since := "2025-01-24")]
theorem MeasureTheory.mem_lpMeasSubgroup_iff_aeStronglyMeasurable' {α : Type u_1} {F : Type u_2} {p : ENNReal} [NormedAddCommGroup F] {m m0 : MeasurableSpace α} {μ : Measure α} {f : (Lp F p μ)} :
f lpMeasSubgroup F m p μ AEStronglyMeasurable (↑f) μ

Alias of MeasureTheory.mem_lpMeasSubgroup_iff_aestronglyMeasurable.

@[deprecated MeasureTheory.mem_lpMeasSubgroup_iff_aestronglyMeasurable (since := "2025-04-09")]
theorem MeasureTheory.mem_lpMeasSubgroup_iff_aeStronglyMeasurable {α : Type u_1} {F : Type u_2} {p : ENNReal} [NormedAddCommGroup F] {m m0 : MeasurableSpace α} {μ : Measure α} {f : (Lp F p μ)} :
f lpMeasSubgroup F m p μ AEStronglyMeasurable (↑f) μ

Alias of MeasureTheory.mem_lpMeasSubgroup_iff_aestronglyMeasurable.

theorem MeasureTheory.mem_lpMeas_iff_aestronglyMeasurable {α : Type u_1} {F : Type u_2} {𝕜 : Type u_3} {p : ENNReal} [RCLike 𝕜] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {m m0 : MeasurableSpace α} {μ : Measure α} {f : (Lp F p μ)} :
f lpMeas F 𝕜 m p μ AEStronglyMeasurable (↑f) μ
@[deprecated MeasureTheory.mem_lpMeas_iff_aestronglyMeasurable (since := "2025-01-24")]
theorem MeasureTheory.mem_lpMeas_iff_aeStronglyMeasurable' {α : Type u_1} {F : Type u_2} {𝕜 : Type u_3} {p : ENNReal} [RCLike 𝕜] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {m m0 : MeasurableSpace α} {μ : Measure α} {f : (Lp F p μ)} :
f lpMeas F 𝕜 m p μ AEStronglyMeasurable (↑f) μ

Alias of MeasureTheory.mem_lpMeas_iff_aestronglyMeasurable.

@[deprecated MeasureTheory.mem_lpMeas_iff_aestronglyMeasurable (since := "2025-04-09")]
theorem MeasureTheory.mem_lpMeas_iff_aeStronglyMeasurable {α : Type u_1} {F : Type u_2} {𝕜 : Type u_3} {p : ENNReal} [RCLike 𝕜] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {m m0 : MeasurableSpace α} {μ : Measure α} {f : (Lp F p μ)} :
f lpMeas F 𝕜 m p μ AEStronglyMeasurable (↑f) μ

Alias of MeasureTheory.mem_lpMeas_iff_aestronglyMeasurable.

theorem MeasureTheory.lpMeas.aestronglyMeasurable {α : Type u_1} {F : Type u_2} {𝕜 : Type u_3} {p : ENNReal} [RCLike 𝕜] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {m x✝ : MeasurableSpace α} {μ : Measure α} (f : (lpMeas F 𝕜 m p μ)) :
AEStronglyMeasurable (↑f) μ
@[deprecated MeasureTheory.lpMeas.aestronglyMeasurable (since := "2025-01-24")]
theorem MeasureTheory.lpMeas.aeStronglyMeasurable' {α : Type u_1} {F : Type u_2} {𝕜 : Type u_3} {p : ENNReal} [RCLike 𝕜] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {m x✝ : MeasurableSpace α} {μ : Measure α} (f : (lpMeas F 𝕜 m p μ)) :
AEStronglyMeasurable (↑f) μ

Alias of MeasureTheory.lpMeas.aestronglyMeasurable.

@[deprecated MeasureTheory.lpMeas.aestronglyMeasurable (since := "2025-04-09")]
theorem MeasureTheory.lpMeas.aeStronglyMeasurable {α : Type u_1} {F : Type u_2} {𝕜 : Type u_3} {p : ENNReal} [RCLike 𝕜] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {m x✝ : MeasurableSpace α} {μ : Measure α} (f : (lpMeas F 𝕜 m p μ)) :
AEStronglyMeasurable (↑f) μ

Alias of MeasureTheory.lpMeas.aestronglyMeasurable.

theorem MeasureTheory.mem_lpMeas_self {α : Type u_1} {F : Type u_2} {𝕜 : Type u_3} {p : ENNReal} [RCLike 𝕜] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {m0 : MeasurableSpace α} (μ : Measure α) (f : (Lp F p μ)) :
f lpMeas F 𝕜 m0 p μ
theorem MeasureTheory.mem_lpMeas_indicatorConstLp {α : Type u_1} {F : Type u_2} {𝕜 : Type u_3} {p : ENNReal} [RCLike 𝕜] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {m m0 : MeasurableSpace α} (hm : m m0) {μ : Measure α} {s : Set α} (hs : MeasurableSet s) (hμs : μ s ) {c : F} :
indicatorConstLp p hμs c lpMeas F 𝕜 m p μ

The subspace lpMeas is complete. #

We define an IsometryEquiv between lpMeasSubgroup and the Lp space corresponding to the measure μ.trim hm. As a consequence, the completeness of Lp implies completeness of lpMeasSubgroup (and lpMeas).

theorem MeasureTheory.memLp_trim_of_mem_lpMeasSubgroup {α : Type u_1} {F : Type u_2} {p : ENNReal} [NormedAddCommGroup F] {m m0 : MeasurableSpace α} {μ : Measure α} (hm : m m0) (f : (Lp F p μ)) (hf_meas : f lpMeasSubgroup F m p μ) :
MemLp (Exists.choose ) p (μ.trim hm)

If f belongs to lpMeasSubgroup F m p μ, then the measurable function it is almost everywhere equal to (given by AEMeasurable.mk) belongs to ℒp for the measure μ.trim hm.

@[deprecated MeasureTheory.memLp_trim_of_mem_lpMeasSubgroup (since := "2025-02-21")]
theorem MeasureTheory.memℒp_trim_of_mem_lpMeasSubgroup {α : Type u_1} {F : Type u_2} {p : ENNReal} [NormedAddCommGroup F] {m m0 : MeasurableSpace α} {μ : Measure α} (hm : m m0) (f : (Lp F p μ)) (hf_meas : f lpMeasSubgroup F m p μ) :
MemLp (Exists.choose ) p (μ.trim hm)

Alias of MeasureTheory.memLp_trim_of_mem_lpMeasSubgroup.


If f belongs to lpMeasSubgroup F m p μ, then the measurable function it is almost everywhere equal to (given by AEMeasurable.mk) belongs to ℒp for the measure μ.trim hm.

theorem MeasureTheory.mem_lpMeasSubgroup_toLp_of_trim {α : Type u_1} {F : Type u_2} {p : ENNReal} [NormedAddCommGroup F] {m m0 : MeasurableSpace α} {μ : Measure α} (hm : m m0) (f : (Lp F p (μ.trim hm))) :
MemLp.toLp f lpMeasSubgroup F m p μ

If f belongs to Lp for the measure μ.trim hm, then it belongs to the subgroup lpMeasSubgroup F m p μ.

noncomputable def MeasureTheory.lpMeasSubgroupToLpTrim {α : Type u_1} (F : Type u_2) (p : ENNReal) [NormedAddCommGroup F] {m m0 : MeasurableSpace α} (μ : Measure α) (hm : m m0) (f : (lpMeasSubgroup F m p μ)) :
(Lp F p (μ.trim hm))

Map from lpMeasSubgroup to Lp F p (μ.trim hm).

Equations
noncomputable def MeasureTheory.lpMeasToLpTrim {α : Type u_1} (F : Type u_2) (𝕜 : Type u_3) (p : ENNReal) [RCLike 𝕜] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {m m0 : MeasurableSpace α} (μ : Measure α) (hm : m m0) (f : (lpMeas F 𝕜 m p μ)) :
(Lp F p (μ.trim hm))

Map from lpMeas to Lp F p (μ.trim hm).

Equations
noncomputable def MeasureTheory.lpTrimToLpMeasSubgroup {α : Type u_1} (F : Type u_2) (p : ENNReal) [NormedAddCommGroup F] {m m0 : MeasurableSpace α} (μ : Measure α) (hm : m m0) (f : (Lp F p (μ.trim hm))) :
(lpMeasSubgroup F m p μ)

Map from Lp F p (μ.trim hm) to lpMeasSubgroup, inverse of lpMeasSubgroupToLpTrim.

Equations
noncomputable def MeasureTheory.lpTrimToLpMeas {α : Type u_1} (F : Type u_2) (𝕜 : Type u_3) (p : ENNReal) [RCLike 𝕜] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {m m0 : MeasurableSpace α} (μ : Measure α) (hm : m m0) (f : (Lp F p (μ.trim hm))) :
(lpMeas F 𝕜 m p μ)

Map from Lp F p (μ.trim hm) to lpMeas, inverse of Lp_meas_to_Lp_trim.

Equations
theorem MeasureTheory.lpMeasSubgroupToLpTrim_ae_eq {α : Type u_1} {F : Type u_2} {p : ENNReal} [NormedAddCommGroup F] {m m0 : MeasurableSpace α} {μ : Measure α} (hm : m m0) (f : (lpMeasSubgroup F m p μ)) :
(lpMeasSubgroupToLpTrim F p μ hm f) =ᶠ[ae μ] f
theorem MeasureTheory.lpTrimToLpMeasSubgroup_ae_eq {α : Type u_1} {F : Type u_2} {p : ENNReal} [NormedAddCommGroup F] {m m0 : MeasurableSpace α} {μ : Measure α} (hm : m m0) (f : (Lp F p (μ.trim hm))) :
(lpTrimToLpMeasSubgroup F p μ hm f) =ᶠ[ae μ] f
theorem MeasureTheory.lpMeasToLpTrim_ae_eq {α : Type u_1} {F : Type u_2} {𝕜 : Type u_3} {p : ENNReal} [RCLike 𝕜] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {m m0 : MeasurableSpace α} {μ : Measure α} (hm : m m0) (f : (lpMeas F 𝕜 m p μ)) :
(lpMeasToLpTrim F 𝕜 p μ hm f) =ᶠ[ae μ] f
theorem MeasureTheory.lpTrimToLpMeas_ae_eq {α : Type u_1} {F : Type u_2} {𝕜 : Type u_3} {p : ENNReal} [RCLike 𝕜] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {m m0 : MeasurableSpace α} {μ : Measure α} (hm : m m0) (f : (Lp F p (μ.trim hm))) :
(lpTrimToLpMeas F 𝕜 p μ hm f) =ᶠ[ae μ] f
theorem MeasureTheory.lpMeasSubgroupToLpTrim_add {α : Type u_1} {F : Type u_2} {p : ENNReal} [NormedAddCommGroup F] {m m0 : MeasurableSpace α} {μ : Measure α} (hm : m m0) (f g : (lpMeasSubgroup F m p μ)) :
theorem MeasureTheory.lpMeasSubgroupToLpTrim_neg {α : Type u_1} {F : Type u_2} {p : ENNReal} [NormedAddCommGroup F] {m m0 : MeasurableSpace α} {μ : Measure α} (hm : m m0) (f : (lpMeasSubgroup F m p μ)) :
theorem MeasureTheory.lpMeasSubgroupToLpTrim_sub {α : Type u_1} {F : Type u_2} {p : ENNReal} [NormedAddCommGroup F] {m m0 : MeasurableSpace α} {μ : Measure α} (hm : m m0) (f g : (lpMeasSubgroup F m p μ)) :
theorem MeasureTheory.lpMeasToLpTrim_smul {α : Type u_1} {F : Type u_2} {𝕜 : Type u_3} {p : ENNReal} [RCLike 𝕜] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {m m0 : MeasurableSpace α} {μ : Measure α} (hm : m m0) (c : 𝕜) (f : (lpMeas F 𝕜 m p μ)) :
lpMeasToLpTrim F 𝕜 p μ hm (c f) = c lpMeasToLpTrim F 𝕜 p μ hm f
theorem MeasureTheory.lpMeasSubgroupToLpTrim_norm_map {α : Type u_1} {F : Type u_2} {p : ENNReal} [NormedAddCommGroup F] {m m0 : MeasurableSpace α} {μ : Measure α} [hp : Fact (1 p)] (hm : m m0) (f : (lpMeasSubgroup F m p μ)) :

lpMeasSubgroupToLpTrim preserves the norm.

theorem MeasureTheory.isometry_lpMeasSubgroupToLpTrim {α : Type u_1} {F : Type u_2} {p : ENNReal} [NormedAddCommGroup F] {m m0 : MeasurableSpace α} {μ : Measure α} [hp : Fact (1 p)] (hm : m m0) :
noncomputable def MeasureTheory.lpMeasSubgroupToLpTrimIso {α : Type u_1} (F : Type u_2) (p : ENNReal) [NormedAddCommGroup F] {m m0 : MeasurableSpace α} (μ : Measure α) [Fact (1 p)] (hm : m m0) :
(lpMeasSubgroup F m p μ) ≃ᵢ (Lp F p (μ.trim hm))

lpMeasSubgroup and Lp F p (μ.trim hm) are isometric.

Equations
  • One or more equations did not get rendered due to their size.
noncomputable def MeasureTheory.lpMeasSubgroupToLpMeasIso {α : Type u_1} (F : Type u_2) (𝕜 : Type u_3) (p : ENNReal) [RCLike 𝕜] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {m m0 : MeasurableSpace α} (μ : Measure α) [Fact (1 p)] :
(lpMeasSubgroup F m p μ) ≃ᵢ (lpMeas F 𝕜 m p μ)

lpMeasSubgroup and lpMeas are isometric.

Equations
noncomputable def MeasureTheory.lpMeasToLpTrimLie {α : Type u_1} (F : Type u_2) (𝕜 : Type u_3) (p : ENNReal) [RCLike 𝕜] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {m m0 : MeasurableSpace α} (μ : Measure α) [Fact (1 p)] (hm : m m0) :
(lpMeas F 𝕜 m p μ) ≃ₗᵢ[𝕜] (Lp F p (μ.trim hm))

lpMeas and Lp F p (μ.trim hm) are isometric, with a linear equivalence.

Equations
  • One or more equations did not get rendered due to their size.
instance MeasureTheory.instCompleteSpaceSubtypeAEEqFunMemAddSubgroupLpSubmoduleLpMeasOfFactLeMeasurableSpace {α : Type u_1} {F : Type u_2} {𝕜 : Type u_3} {p : ENNReal} [RCLike 𝕜] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {m m0 : MeasurableSpace α} {μ : Measure α} [hm : Fact (m m0)] [CompleteSpace F] [hp : Fact (1 p)] :
CompleteSpace (lpMeas F 𝕜 m p μ)
theorem MeasureTheory.isComplete_aestronglyMeasurable {α : Type u_1} {F : Type u_2} {p : ENNReal} [NormedAddCommGroup F] {m m0 : MeasurableSpace α} {μ : Measure α} [hp : Fact (1 p)] [CompleteSpace F] (hm : m m0) :
IsComplete {f : (Lp F p μ) | AEStronglyMeasurable (↑f) μ}
@[deprecated MeasureTheory.isComplete_aestronglyMeasurable (since := "2025-04-09")]
theorem MeasureTheory.isComplete_aeStronglyMeasurable' {α : Type u_1} {F : Type u_2} {p : ENNReal} [NormedAddCommGroup F] {m m0 : MeasurableSpace α} {μ : Measure α} [hp : Fact (1 p)] [CompleteSpace F] (hm : m m0) :
IsComplete {f : (Lp F p μ) | AEStronglyMeasurable (↑f) μ}

Alias of MeasureTheory.isComplete_aestronglyMeasurable.

theorem MeasureTheory.isClosed_aestronglyMeasurable {α : Type u_1} {F : Type u_2} {p : ENNReal} [NormedAddCommGroup F] {m m0 : MeasurableSpace α} {μ : Measure α} [Fact (1 p)] [CompleteSpace F] (hm : m m0) :
IsClosed {f : (Lp F p μ) | AEStronglyMeasurable (↑f) μ}
@[deprecated MeasureTheory.isClosed_aestronglyMeasurable (since := "2025-04-09")]
theorem MeasureTheory.isClosed_aeStronglyMeasurable' {α : Type u_1} {F : Type u_2} {p : ENNReal} [NormedAddCommGroup F] {m m0 : MeasurableSpace α} {μ : Measure α} [Fact (1 p)] [CompleteSpace F] (hm : m m0) :
IsClosed {f : (Lp F p μ) | AEStronglyMeasurable (↑f) μ}

Alias of MeasureTheory.isClosed_aestronglyMeasurable.

theorem MeasureTheory.lpMeas.ae_fin_strongly_measurable' {α : Type u_1} {F : Type u_2} {𝕜 : Type u_3} {p : ENNReal} [RCLike 𝕜] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {m m0 : MeasurableSpace α} {μ : Measure α} (hm : m m0) (f : (lpMeas F 𝕜 m p μ)) (hp_ne_zero : p 0) (hp_ne_top : p ) :
∃ (g : αF), FinStronglyMeasurable g (μ.trim hm) f =ᶠ[ae μ] g

We do not get ae_fin_strongly_measurable f (μ.trim hm), since we don't have f =ᵐ[μ.trim hm] Lp_meas_to_Lp_trim F 𝕜 p μ hm f but only the weaker f =ᵐ[μ] Lp_meas_to_Lp_trim F 𝕜 p μ hm f.

theorem MeasureTheory.lpMeasToLpTrimLie_symm_indicator {α : Type u_1} {F : Type u_2} {p : ENNReal} [NormedAddCommGroup F] {m m0 : MeasurableSpace α} [one_le_p : Fact (1 p)] [NormedSpace F] {hm : m m0} {s : Set α} {μ : Measure α} (hs : MeasurableSet s) (hμs : (μ.trim hm) s ) (c : F) :
((lpMeasToLpTrimLie F p μ hm).symm (indicatorConstLp p hs hμs c)) = indicatorConstLp p c

When applying the inverse of lpMeasToLpTrimLie (which takes a function in the Lp space of the sub-sigma algebra and returns its version in the larger Lp space) to an indicator of the sub-sigma-algebra, we obtain an indicator in the Lp space of the larger sigma-algebra.

theorem MeasureTheory.lpMeasToLpTrimLie_symm_toLp {α : Type u_1} {F : Type u_2} {p : ENNReal} [NormedAddCommGroup F] {m m0 : MeasurableSpace α} {μ : Measure α} [one_le_p : Fact (1 p)] [NormedSpace F] (hm : m m0) (f : αF) (hf : MemLp f p (μ.trim hm)) :
((lpMeasToLpTrimLie F p μ hm).symm (MemLp.toLp f hf)) = MemLp.toLp f
theorem MeasureTheory.Lp.induction_stronglyMeasurable_aux {α : Type u_1} {F : Type u_2} {p : ENNReal} [NormedAddCommGroup F] {m m0 : MeasurableSpace α} {μ : Measure α} [Fact (1 p)] [NormedSpace F] (hm : m m0) (hp_ne_top : p ) (P : (Lp F p μ)Prop) (h_ind : ∀ (c : F) {s : Set α} (hs : MeasurableSet s) (hμs : μ s < ), P (simpleFunc.indicatorConst p c)) (h_add : ∀ ⦃f g : αF⦄ (hf : MemLp f p μ) (hg : MemLp g p μ), AEStronglyMeasurable f μAEStronglyMeasurable g μDisjoint (Function.support f) (Function.support g)P (MemLp.toLp f hf)P (MemLp.toLp g hg)P (MemLp.toLp f hf + MemLp.toLp g hg)) (h_closed : IsClosed {f : (lpMeas F m p μ) | P f}) (f : (Lp F p μ)) :
AEStronglyMeasurable (↑f) μP f

Auxiliary lemma for Lp.induction_stronglyMeasurable.

theorem MeasureTheory.Lp.induction_stronglyMeasurable {α : Type u_1} {F : Type u_2} {p : ENNReal} [NormedAddCommGroup F] {m m0 : MeasurableSpace α} {μ : Measure α} [Fact (1 p)] [NormedSpace F] (hm : m m0) (hp_ne_top : p ) (P : (Lp F p μ)Prop) (h_ind : ∀ (c : F) {s : Set α} (hs : MeasurableSet s) (hμs : μ s < ), P (simpleFunc.indicatorConst p c)) (h_add : ∀ ⦃f g : αF⦄ (hf : MemLp f p μ) (hg : MemLp g p μ), StronglyMeasurable fStronglyMeasurable gDisjoint (Function.support f) (Function.support g)P (MemLp.toLp f hf)P (MemLp.toLp g hg)P (MemLp.toLp f hf + MemLp.toLp g hg)) (h_closed : IsClosed {f : (lpMeas F m p μ) | P f}) (f : (Lp F p μ)) :
AEStronglyMeasurable (↑f) μP f

To prove something for an Lp function a.e. strongly measurable with respect to a sub-σ-algebra m in a normed space, it suffices to show that

  • the property holds for (multiples of) characteristic functions which are measurable w.r.t. m;
  • is closed under addition;
  • the set of functions in Lp strongly measurable w.r.t. m for which the property holds is closed.
theorem MeasureTheory.MemLp.induction_stronglyMeasurable {α : Type u_1} {F : Type u_2} {p : ENNReal} [NormedAddCommGroup F] {m m0 : MeasurableSpace α} {μ : Measure α} [Fact (1 p)] [NormedSpace F] (hm : m m0) (hp_ne_top : p ) (P : (αF)Prop) (h_ind : ∀ (c : F) ⦃s : Set α⦄, MeasurableSet sμ s < P (s.indicator fun (x : α) => c)) (h_add : ∀ ⦃f g : αF⦄, Disjoint (Function.support f) (Function.support g)MemLp f p μMemLp g p μStronglyMeasurable fStronglyMeasurable gP fP gP (f + g)) (h_closed : IsClosed {f : (lpMeas F m p μ) | P f}) (h_ae : ∀ ⦃f g : αF⦄, f =ᶠ[ae μ] gMemLp f p μP fP g) f : αF :
MemLp f p μAEStronglyMeasurable f μP f

To prove something for an arbitrary MemLp function a.e. strongly measurable with respect to a sub-σ-algebra m in a normed space, it suffices to show that

  • the property holds for (multiples of) characteristic functions which are measurable w.r.t. m;
  • is closed under addition;
  • the set of functions in the Lᵖ space strongly measurable w.r.t. m for which the property holds is closed.
  • the property is closed under the almost-everywhere equal relation.
@[deprecated MeasureTheory.MemLp.induction_stronglyMeasurable (since := "2025-02-21")]
theorem MeasureTheory.Memℒp.induction_stronglyMeasurable {α : Type u_1} {F : Type u_2} {p : ENNReal} [NormedAddCommGroup F] {m m0 : MeasurableSpace α} {μ : Measure α} [Fact (1 p)] [NormedSpace F] (hm : m m0) (hp_ne_top : p ) (P : (αF)Prop) (h_ind : ∀ (c : F) ⦃s : Set α⦄, MeasurableSet sμ s < P (s.indicator fun (x : α) => c)) (h_add : ∀ ⦃f g : αF⦄, Disjoint (Function.support f) (Function.support g)MemLp f p μMemLp g p μStronglyMeasurable fStronglyMeasurable gP fP gP (f + g)) (h_closed : IsClosed {f : (lpMeas F m p μ) | P f}) (h_ae : ∀ ⦃f g : αF⦄, f =ᶠ[ae μ] gMemLp f p μP fP g) f : αF :
MemLp f p μAEStronglyMeasurable f μP f

Alias of MeasureTheory.MemLp.induction_stronglyMeasurable.


To prove something for an arbitrary MemLp function a.e. strongly measurable with respect to a sub-σ-algebra m in a normed space, it suffices to show that

  • the property holds for (multiples of) characteristic functions which are measurable w.r.t. m;
  • is closed under addition;
  • the set of functions in the Lᵖ space strongly measurable w.r.t. m for which the property holds is closed.
  • the property is closed under the almost-everywhere equal relation.