fact stringlengths 6 3.84k | type stringclasses 11
values | library stringclasses 32
values | imports listlengths 1 14 | filename stringlengths 20 95 | symbolic_name stringlengths 1 90 | docstring stringlengths 7 20k ⌀ |
|---|---|---|---|---|---|---|
condCDF_nonneg (ρ : Measure (α × ℝ)) (a : α) (r : ℝ) : 0 ≤ condCDF ρ a r :=
stieltjesOfMeasurableRat_nonneg _ a r | theorem | Probability | [
"Mathlib.MeasureTheory.Measure.Decomposition.RadonNikodym",
"Mathlib.MeasureTheory.Measure.Prod",
"Mathlib.Probability.Kernel.Disintegration.CDFToKernel"
] | Mathlib/Probability/Kernel/Disintegration/CondCDF.lean | condCDF_nonneg | The conditional cdf is non-negative for all `a : α`. |
condCDF_le_one (ρ : Measure (α × ℝ)) (a : α) (x : ℝ) : condCDF ρ a x ≤ 1 :=
stieltjesOfMeasurableRat_le_one _ _ _ | theorem | Probability | [
"Mathlib.MeasureTheory.Measure.Decomposition.RadonNikodym",
"Mathlib.MeasureTheory.Measure.Prod",
"Mathlib.Probability.Kernel.Disintegration.CDFToKernel"
] | Mathlib/Probability/Kernel/Disintegration/CondCDF.lean | condCDF_le_one | The conditional cdf is lower or equal to 1 for all `a : α`. |
tendsto_condCDF_atBot (ρ : Measure (α × ℝ)) (a : α) :
Tendsto (condCDF ρ a) atBot (𝓝 0) := tendsto_stieltjesOfMeasurableRat_atBot _ _ | theorem | Probability | [
"Mathlib.MeasureTheory.Measure.Decomposition.RadonNikodym",
"Mathlib.MeasureTheory.Measure.Prod",
"Mathlib.Probability.Kernel.Disintegration.CDFToKernel"
] | Mathlib/Probability/Kernel/Disintegration/CondCDF.lean | tendsto_condCDF_atBot | The conditional cdf tends to 0 at -∞ for all `a : α`. |
tendsto_condCDF_atTop (ρ : Measure (α × ℝ)) (a : α) :
Tendsto (condCDF ρ a) atTop (𝓝 1) := tendsto_stieltjesOfMeasurableRat_atTop _ _ | theorem | Probability | [
"Mathlib.MeasureTheory.Measure.Decomposition.RadonNikodym",
"Mathlib.MeasureTheory.Measure.Prod",
"Mathlib.Probability.Kernel.Disintegration.CDFToKernel"
] | Mathlib/Probability/Kernel/Disintegration/CondCDF.lean | tendsto_condCDF_atTop | The conditional cdf tends to 1 at +∞ for all `a : α`. |
condCDF_ae_eq (ρ : Measure (α × ℝ)) [IsFiniteMeasure ρ] (r : ℚ) :
(fun a ↦ condCDF ρ a r) =ᵐ[ρ.fst] fun a ↦ (preCDF ρ r a).toReal := by
simp_rw [condCDF_eq_stieltjesOfMeasurableRat_unit_prod ρ]
exact stieltjesOfMeasurableRat_ae_eq (isRatCondKernelCDF_preCDF ρ) () r | theorem | Probability | [
"Mathlib.MeasureTheory.Measure.Decomposition.RadonNikodym",
"Mathlib.MeasureTheory.Measure.Prod",
"Mathlib.Probability.Kernel.Disintegration.CDFToKernel"
] | Mathlib/Probability/Kernel/Disintegration/CondCDF.lean | condCDF_ae_eq | null |
ofReal_condCDF_ae_eq (ρ : Measure (α × ℝ)) [IsFiniteMeasure ρ] (r : ℚ) :
(fun a ↦ ENNReal.ofReal (condCDF ρ a r)) =ᵐ[ρ.fst] preCDF ρ r := by
filter_upwards [condCDF_ae_eq ρ r, preCDF_le_one ρ] with a ha ha_le_one
rw [ha, ENNReal.ofReal_toReal]
exact ((ha_le_one r).trans_lt ENNReal.one_lt_top).ne | theorem | Probability | [
"Mathlib.MeasureTheory.Measure.Decomposition.RadonNikodym",
"Mathlib.MeasureTheory.Measure.Prod",
"Mathlib.Probability.Kernel.Disintegration.CDFToKernel"
] | Mathlib/Probability/Kernel/Disintegration/CondCDF.lean | ofReal_condCDF_ae_eq | null |
measurable_condCDF (ρ : Measure (α × ℝ)) (x : ℝ) : Measurable fun a ↦ condCDF ρ a x :=
measurable_stieltjesOfMeasurableRat _ _ | theorem | Probability | [
"Mathlib.MeasureTheory.Measure.Decomposition.RadonNikodym",
"Mathlib.MeasureTheory.Measure.Prod",
"Mathlib.Probability.Kernel.Disintegration.CDFToKernel"
] | Mathlib/Probability/Kernel/Disintegration/CondCDF.lean | measurable_condCDF | The conditional cdf is a measurable function of `a : α` for all `x : ℝ`. |
stronglyMeasurable_condCDF (ρ : Measure (α × ℝ)) (x : ℝ) :
StronglyMeasurable fun a ↦ condCDF ρ a x := stronglyMeasurable_stieltjesOfMeasurableRat _ _ | theorem | Probability | [
"Mathlib.MeasureTheory.Measure.Decomposition.RadonNikodym",
"Mathlib.MeasureTheory.Measure.Prod",
"Mathlib.Probability.Kernel.Disintegration.CDFToKernel"
] | Mathlib/Probability/Kernel/Disintegration/CondCDF.lean | stronglyMeasurable_condCDF | The conditional cdf is a strongly measurable function of `a : α` for all `x : ℝ`. |
setLIntegral_condCDF (ρ : Measure (α × ℝ)) [IsFiniteMeasure ρ] (x : ℝ) {s : Set α}
(hs : MeasurableSet s) :
∫⁻ a in s, ENNReal.ofReal (condCDF ρ a x) ∂ρ.fst = ρ (s ×ˢ Iic x) :=
(isCondKernelCDF_condCDF ρ).setLIntegral () hs x | theorem | Probability | [
"Mathlib.MeasureTheory.Measure.Decomposition.RadonNikodym",
"Mathlib.MeasureTheory.Measure.Prod",
"Mathlib.Probability.Kernel.Disintegration.CDFToKernel"
] | Mathlib/Probability/Kernel/Disintegration/CondCDF.lean | setLIntegral_condCDF | null |
lintegral_condCDF (ρ : Measure (α × ℝ)) [IsFiniteMeasure ρ] (x : ℝ) :
∫⁻ a, ENNReal.ofReal (condCDF ρ a x) ∂ρ.fst = ρ (univ ×ˢ Iic x) :=
(isCondKernelCDF_condCDF ρ).lintegral () x | theorem | Probability | [
"Mathlib.MeasureTheory.Measure.Decomposition.RadonNikodym",
"Mathlib.MeasureTheory.Measure.Prod",
"Mathlib.Probability.Kernel.Disintegration.CDFToKernel"
] | Mathlib/Probability/Kernel/Disintegration/CondCDF.lean | lintegral_condCDF | null |
integrable_condCDF (ρ : Measure (α × ℝ)) [IsFiniteMeasure ρ] (x : ℝ) :
Integrable (fun a ↦ condCDF ρ a x) ρ.fst :=
(isCondKernelCDF_condCDF ρ).integrable () x | theorem | Probability | [
"Mathlib.MeasureTheory.Measure.Decomposition.RadonNikodym",
"Mathlib.MeasureTheory.Measure.Prod",
"Mathlib.Probability.Kernel.Disintegration.CDFToKernel"
] | Mathlib/Probability/Kernel/Disintegration/CondCDF.lean | integrable_condCDF | null |
setIntegral_condCDF (ρ : Measure (α × ℝ)) [IsFiniteMeasure ρ] (x : ℝ) {s : Set α}
(hs : MeasurableSet s) : ∫ a in s, condCDF ρ a x ∂ρ.fst = ρ.real (s ×ˢ Iic x) :=
(isCondKernelCDF_condCDF ρ).setIntegral () hs x | theorem | Probability | [
"Mathlib.MeasureTheory.Measure.Decomposition.RadonNikodym",
"Mathlib.MeasureTheory.Measure.Prod",
"Mathlib.Probability.Kernel.Disintegration.CDFToKernel"
] | Mathlib/Probability/Kernel/Disintegration/CondCDF.lean | setIntegral_condCDF | null |
integral_condCDF (ρ : Measure (α × ℝ)) [IsFiniteMeasure ρ] (x : ℝ) :
∫ a, condCDF ρ a x ∂ρ.fst = ρ.real (univ ×ˢ Iic x) :=
(isCondKernelCDF_condCDF ρ).integral () x | theorem | Probability | [
"Mathlib.MeasureTheory.Measure.Decomposition.RadonNikodym",
"Mathlib.MeasureTheory.Measure.Prod",
"Mathlib.Probability.Kernel.Disintegration.CDFToKernel"
] | Mathlib/Probability/Kernel/Disintegration/CondCDF.lean | integral_condCDF | null |
measure_condCDF_Iic (ρ : Measure (α × ℝ)) (a : α) (x : ℝ) :
(condCDF ρ a).measure (Iic x) = ENNReal.ofReal (condCDF ρ a x) := by
rw [← sub_zero (condCDF ρ a x)]
exact (condCDF ρ a).measure_Iic (tendsto_condCDF_atBot ρ a) _ | theorem | Probability | [
"Mathlib.MeasureTheory.Measure.Decomposition.RadonNikodym",
"Mathlib.MeasureTheory.Measure.Prod",
"Mathlib.Probability.Kernel.Disintegration.CDFToKernel"
] | Mathlib/Probability/Kernel/Disintegration/CondCDF.lean | measure_condCDF_Iic | null |
measure_condCDF_univ (ρ : Measure (α × ℝ)) (a : α) : (condCDF ρ a).measure univ = 1 := by
rw [← ENNReal.ofReal_one, ← sub_zero (1 : ℝ)]
exact StieltjesFunction.measure_univ _ (tendsto_condCDF_atBot ρ a) (tendsto_condCDF_atTop ρ a) | theorem | Probability | [
"Mathlib.MeasureTheory.Measure.Decomposition.RadonNikodym",
"Mathlib.MeasureTheory.Measure.Prod",
"Mathlib.Probability.Kernel.Disintegration.CDFToKernel"
] | Mathlib/Probability/Kernel/Disintegration/CondCDF.lean | measure_condCDF_univ | null |
instIsProbabilityMeasureCondCDF (ρ : Measure (α × ℝ)) (a : α) :
IsProbabilityMeasure (condCDF ρ a).measure :=
⟨measure_condCDF_univ ρ a⟩ | instance | Probability | [
"Mathlib.MeasureTheory.Measure.Decomposition.RadonNikodym",
"Mathlib.MeasureTheory.Measure.Prod",
"Mathlib.Probability.Kernel.Disintegration.CDFToKernel"
] | Mathlib/Probability/Kernel/Disintegration/CondCDF.lean | instIsProbabilityMeasureCondCDF | null |
measurable_measure_condCDF (ρ : Measure (α × ℝ)) :
Measurable fun a => (condCDF ρ a).measure :=
.measure_of_isPiSystem_of_isProbabilityMeasure (borel_eq_generateFrom_Iic ℝ) isPiSystem_Iic <| by
simp_rw [forall_mem_range, measure_condCDF_Iic]
exact fun u ↦ (measurable_condCDF ρ u).ennreal_ofReal | theorem | Probability | [
"Mathlib.MeasureTheory.Measure.Decomposition.RadonNikodym",
"Mathlib.MeasureTheory.Measure.Prod",
"Mathlib.Probability.Kernel.Disintegration.CDFToKernel"
] | Mathlib/Probability/Kernel/Disintegration/CondCDF.lean | measurable_measure_condCDF | The function `a ↦ (condCDF ρ a).measure` is measurable. |
density' (κ : Kernel α (γ × β)) (ν : kernel a γ) (a : α) (x : γ) (s : Set β) : ℝ :=
(((κ a).restrict (univ ×ˢ s)).fst.rnDeriv (ν a) x).toReal
```
However, we can't turn those functions for each `a` into a measurable function of the pair `(a, x)`.
In order to obtain measurability through countability, we use the fact that the measurable space `γ`
is countably generated. For each `n : ℕ`, we define (in the file
`Mathlib/Probability/Process/PartitionFiltration.lean`) a finite partition of `γ`, such that those
partitions are finer as `n` grows, and the σ-algebra generated by the union of all partitions is the
σ-algebra of `γ`. For `x : γ`, `countablePartitionSet n x` denotes the set in the partition such
that `x ∈ countablePartitionSet n x`.
For a given `n`, the function `densityProcess κ ν n : α → γ → Set β → ℝ` defined by
`fun a x s ↦ (κ a (countablePartitionSet n x ×ˢ s) / ν a (countablePartitionSet n x)).toReal` has
the desired property that `∫ x in A, densityProcess κ ν n a x s ∂(ν a) = (κ a (A ×ˢ s)).toReal` for
all `A` in the σ-algebra generated by the partition at scale `n` and is measurable in `(a, x)`.
`countableFiltration γ` is the filtration of those σ-algebras for all `n : ℕ`.
The functions `densityProcess κ ν n` described here are a bounded `ν`-martingale for the filtration
`countableFiltration γ`. By Doob's martingale L1 convergence theorem, that martingale converges to
a limit, which has a product-measurable version and satisfies the integral equality for all `A` in
`⨆ n, countableFiltration γ n`. Finally, the partitions were chosen such that that supremum is equal
to the σ-algebra on `γ`, hence the equality holds for all measurable sets.
We have obtained the desired density function. | def | Probability | [
"Mathlib.Probability.Kernel.Composition.MapComap",
"Mathlib.Probability.Martingale.Convergence",
"Mathlib.Probability.Process.PartitionFiltration"
] | Mathlib/Probability/Kernel/Disintegration/Density.lean | density' | null |
noncomputable
densityProcess (κ : Kernel α (γ × β)) (ν : Kernel α γ) (n : ℕ) (a : α) (x : γ) (s : Set β) :
ℝ :=
(κ a (countablePartitionSet n x ×ˢ s) / ν a (countablePartitionSet n x)).toReal | def | Probability | [
"Mathlib.Probability.Kernel.Composition.MapComap",
"Mathlib.Probability.Martingale.Convergence",
"Mathlib.Probability.Process.PartitionFiltration"
] | Mathlib/Probability/Kernel/Disintegration/Density.lean | densityProcess | An `ℕ`-indexed martingale that is a density for `κ` with respect to `ν` on the sets in
`countablePartition γ n`. Used to define its limit `ProbabilityTheory.Kernel.density`, which is
a density for those kernels for all measurable sets. |
densityProcess_def (κ : Kernel α (γ × β)) (ν : Kernel α γ) (n : ℕ) (a : α) (s : Set β) :
(fun t ↦ densityProcess κ ν n a t s)
= fun t ↦ (κ a (countablePartitionSet n t ×ˢ s) / ν a (countablePartitionSet n t)).toReal :=
rfl | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.MapComap",
"Mathlib.Probability.Martingale.Convergence",
"Mathlib.Probability.Process.PartitionFiltration"
] | Mathlib/Probability/Kernel/Disintegration/Density.lean | densityProcess_def | null |
measurable_densityProcess_countableFiltration_aux (κ : Kernel α (γ × β)) (ν : Kernel α γ)
(n : ℕ) {s : Set β} (hs : MeasurableSet s) :
Measurable[mα.prod (countableFiltration γ n)] (fun (p : α × γ) ↦
κ p.1 (countablePartitionSet n p.2 ×ˢ s) / ν p.1 (countablePartitionSet n p.2)) := by
change Measurable[mα.prod (countableFiltration γ n)]
((fun (p : α × countablePartition γ n) ↦ κ p.1 (↑p.2 ×ˢ s) / ν p.1 p.2)
∘ (fun (p : α × γ) ↦ (p.1, ⟨countablePartitionSet n p.2, countablePartitionSet_mem n p.2⟩)))
have h1 : @Measurable _ _ (mα.prod ⊤) _
(fun p : α × countablePartition γ n ↦ κ p.1 (↑p.2 ×ˢ s) / ν p.1 p.2) := by
refine Measurable.div ?_ ?_
· refine measurable_from_prod_countable_left (fun t ↦ ?_)
exact Kernel.measurable_coe _ ((measurableSet_countablePartition _ t.prop).prod hs)
· refine measurable_from_prod_countable_left ?_
rintro ⟨t, ht⟩
exact Kernel.measurable_coe _ (measurableSet_countablePartition _ ht)
refine h1.comp (measurable_fst.prodMk ?_)
change @Measurable (α × γ) (countablePartition γ n) (mα.prod (countableFiltration γ n)) ⊤
((fun c ↦ ⟨countablePartitionSet n c, countablePartitionSet_mem n c⟩) ∘ (fun p : α × γ ↦ p.2))
exact (measurable_countablePartitionSet_subtype n ⊤).comp measurable_snd | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.MapComap",
"Mathlib.Probability.Martingale.Convergence",
"Mathlib.Probability.Process.PartitionFiltration"
] | Mathlib/Probability/Kernel/Disintegration/Density.lean | measurable_densityProcess_countableFiltration_aux | null |
measurable_densityProcess_aux (κ : Kernel α (γ × β)) (ν : Kernel α γ) (n : ℕ)
{s : Set β} (hs : MeasurableSet s) :
Measurable (fun (p : α × γ) ↦
κ p.1 (countablePartitionSet n p.2 ×ˢ s) / ν p.1 (countablePartitionSet n p.2)) := by
refine Measurable.mono (measurable_densityProcess_countableFiltration_aux κ ν n hs) ?_ le_rfl
exact sup_le_sup le_rfl (comap_mono ((countableFiltration γ).le _)) | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.MapComap",
"Mathlib.Probability.Martingale.Convergence",
"Mathlib.Probability.Process.PartitionFiltration"
] | Mathlib/Probability/Kernel/Disintegration/Density.lean | measurable_densityProcess_aux | null |
measurable_densityProcess (κ : Kernel α (γ × β)) (ν : Kernel α γ) (n : ℕ)
{s : Set β} (hs : MeasurableSet s) :
Measurable (fun (p : α × γ) ↦ densityProcess κ ν n p.1 p.2 s) :=
(measurable_densityProcess_aux κ ν n hs).ennreal_toReal | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.MapComap",
"Mathlib.Probability.Martingale.Convergence",
"Mathlib.Probability.Process.PartitionFiltration"
] | Mathlib/Probability/Kernel/Disintegration/Density.lean | measurable_densityProcess | null |
measurable_densityProcess_left (κ : Kernel α (γ × β)) (ν : Kernel α γ) (n : ℕ)
(x : γ) {s : Set β} (hs : MeasurableSet s) :
Measurable (fun a ↦ densityProcess κ ν n a x s) :=
((measurable_densityProcess κ ν n hs).comp (measurable_id.prodMk measurable_const):) | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.MapComap",
"Mathlib.Probability.Martingale.Convergence",
"Mathlib.Probability.Process.PartitionFiltration"
] | Mathlib/Probability/Kernel/Disintegration/Density.lean | measurable_densityProcess_left | null |
measurable_densityProcess_right (κ : Kernel α (γ × β)) (ν : Kernel α γ) (n : ℕ)
{s : Set β} (a : α) (hs : MeasurableSet s) :
Measurable (fun x ↦ densityProcess κ ν n a x s) :=
((measurable_densityProcess κ ν n hs).comp (measurable_const.prodMk measurable_id):) | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.MapComap",
"Mathlib.Probability.Martingale.Convergence",
"Mathlib.Probability.Process.PartitionFiltration"
] | Mathlib/Probability/Kernel/Disintegration/Density.lean | measurable_densityProcess_right | null |
measurable_countableFiltration_densityProcess (κ : Kernel α (γ × β)) (ν : Kernel α γ) (n : ℕ)
(a : α) {s : Set β} (hs : MeasurableSet s) :
Measurable[countableFiltration γ n] (fun x ↦ densityProcess κ ν n a x s) := by
refine @Measurable.ennreal_toReal _ (countableFiltration γ n) _ ?_
exact (measurable_densityProcess_countableFiltration_aux κ ν n hs).comp measurable_prodMk_left | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.MapComap",
"Mathlib.Probability.Martingale.Convergence",
"Mathlib.Probability.Process.PartitionFiltration"
] | Mathlib/Probability/Kernel/Disintegration/Density.lean | measurable_countableFiltration_densityProcess | null |
stronglyMeasurable_countableFiltration_densityProcess (κ : Kernel α (γ × β)) (ν : Kernel α γ)
(n : ℕ) (a : α) {s : Set β} (hs : MeasurableSet s) :
StronglyMeasurable[countableFiltration γ n] (fun x ↦ densityProcess κ ν n a x s) :=
(measurable_countableFiltration_densityProcess κ ν n a hs).stronglyMeasurable | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.MapComap",
"Mathlib.Probability.Martingale.Convergence",
"Mathlib.Probability.Process.PartitionFiltration"
] | Mathlib/Probability/Kernel/Disintegration/Density.lean | stronglyMeasurable_countableFiltration_densityProcess | null |
adapted_densityProcess (κ : Kernel α (γ × β)) (ν : Kernel α γ) (a : α)
{s : Set β} (hs : MeasurableSet s) :
Adapted (countableFiltration γ) (fun n x ↦ densityProcess κ ν n a x s) :=
fun n ↦ stronglyMeasurable_countableFiltration_densityProcess κ ν n a hs | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.MapComap",
"Mathlib.Probability.Martingale.Convergence",
"Mathlib.Probability.Process.PartitionFiltration"
] | Mathlib/Probability/Kernel/Disintegration/Density.lean | adapted_densityProcess | null |
densityProcess_nonneg (κ : Kernel α (γ × β)) (ν : Kernel α γ) (n : ℕ)
(a : α) (x : γ) (s : Set β) :
0 ≤ densityProcess κ ν n a x s :=
ENNReal.toReal_nonneg | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.MapComap",
"Mathlib.Probability.Martingale.Convergence",
"Mathlib.Probability.Process.PartitionFiltration"
] | Mathlib/Probability/Kernel/Disintegration/Density.lean | densityProcess_nonneg | null |
meas_countablePartitionSet_le_of_fst_le (hκν : fst κ ≤ ν) (n : ℕ) (a : α) (x : γ)
(s : Set β) :
κ a (countablePartitionSet n x ×ˢ s) ≤ ν a (countablePartitionSet n x) := by
calc κ a (countablePartitionSet n x ×ˢ s)
≤ fst κ a (countablePartitionSet n x) := by
rw [fst_apply' _ _ (measurableSet_countablePartitionSet _ _)]
refine measure_mono (fun x ↦ ?_)
simp only [mem_prod, mem_setOf_eq, and_imp]
exact fun h _ ↦ h
_ ≤ ν a (countablePartitionSet n x) := hκν a _ | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.MapComap",
"Mathlib.Probability.Martingale.Convergence",
"Mathlib.Probability.Process.PartitionFiltration"
] | Mathlib/Probability/Kernel/Disintegration/Density.lean | meas_countablePartitionSet_le_of_fst_le | null |
densityProcess_le_one (hκν : fst κ ≤ ν) (n : ℕ) (a : α) (x : γ) (s : Set β) :
densityProcess κ ν n a x s ≤ 1 := by
refine ENNReal.toReal_le_of_le_ofReal zero_le_one (ENNReal.div_le_of_le_mul ?_)
rw [ENNReal.ofReal_one, one_mul]
exact meas_countablePartitionSet_le_of_fst_le hκν n a x s | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.MapComap",
"Mathlib.Probability.Martingale.Convergence",
"Mathlib.Probability.Process.PartitionFiltration"
] | Mathlib/Probability/Kernel/Disintegration/Density.lean | densityProcess_le_one | null |
eLpNorm_densityProcess_le (hκν : fst κ ≤ ν) (n : ℕ) (a : α) (s : Set β) :
eLpNorm (fun x ↦ densityProcess κ ν n a x s) 1 (ν a) ≤ ν a univ := by
refine (eLpNorm_le_of_ae_bound (C := 1) (ae_of_all _ (fun x ↦ ?_))).trans ?_
· simp only [Real.norm_eq_abs, abs_of_nonneg (densityProcess_nonneg κ ν n a x s),
densityProcess_le_one hκν n a x s]
· simp | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.MapComap",
"Mathlib.Probability.Martingale.Convergence",
"Mathlib.Probability.Process.PartitionFiltration"
] | Mathlib/Probability/Kernel/Disintegration/Density.lean | eLpNorm_densityProcess_le | null |
integrable_densityProcess (hκν : fst κ ≤ ν) [IsFiniteKernel ν] (n : ℕ)
(a : α) {s : Set β} (hs : MeasurableSet s) :
Integrable (fun x ↦ densityProcess κ ν n a x s) (ν a) := by
rw [← memLp_one_iff_integrable]
refine ⟨Measurable.aestronglyMeasurable ?_, ?_⟩
· exact measurable_densityProcess_right κ ν n a hs
· exact (eLpNorm_densityProcess_le hκν n a s).trans_lt (measure_lt_top _ _) | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.MapComap",
"Mathlib.Probability.Martingale.Convergence",
"Mathlib.Probability.Process.PartitionFiltration"
] | Mathlib/Probability/Kernel/Disintegration/Density.lean | integrable_densityProcess | null |
setIntegral_densityProcess_of_mem (hκν : fst κ ≤ ν) [hν : IsFiniteKernel ν]
(n : ℕ) (a : α) {s : Set β} (hs : MeasurableSet s) {u : Set γ}
(hu : u ∈ countablePartition γ n) :
∫ x in u, densityProcess κ ν n a x s ∂(ν a) = (κ a).real (u ×ˢ s) := by
have : IsFiniteKernel κ := isFiniteKernel_of_isFiniteKernel_fst (h := isFiniteKernel_of_le hκν)
have hu_meas : MeasurableSet u := measurableSet_countablePartition n hu
simp_rw [densityProcess]
rw [integral_toReal]
rotate_left
· refine Measurable.aemeasurable ?_
change Measurable ((fun (p : α × _) ↦ κ p.1 (countablePartitionSet n p.2 ×ˢ s)
/ ν p.1 (countablePartitionSet n p.2)) ∘ (fun x ↦ (a, x)))
exact (measurable_densityProcess_aux κ ν n hs).comp measurable_prodMk_left
· refine ae_of_all _ (fun x ↦ ?_)
by_cases h0 : ν a (countablePartitionSet n x) = 0
· suffices κ a (countablePartitionSet n x ×ˢ s) = 0 by simp [h0, this]
have h0' : fst κ a (countablePartitionSet n x) = 0 :=
le_antisymm ((hκν a _).trans h0.le) zero_le'
rw [fst_apply' _ _ (measurableSet_countablePartitionSet _ _)] at h0'
refine measure_mono_null (fun x ↦ ?_) h0'
simp only [mem_prod, mem_setOf_eq, and_imp]
exact fun h _ ↦ h
· finiteness
congr
have : ∫⁻ x in u, κ a (countablePartitionSet n x ×ˢ s) / ν a (countablePartitionSet n x) ∂(ν a)
= ∫⁻ _ in u, κ a (u ×ˢ s) / ν a u ∂(ν a) := by
refine setLIntegral_congr_fun hu_meas (fun t ht ↦ ?_)
rw [countablePartitionSet_of_mem hu ht]
rw [this]
simp only [MeasureTheory.lintegral_const, MeasurableSet.univ, Measure.restrict_apply, univ_inter]
by_cases h0 : ν a u = 0
· simp only [h0, mul_zero]
have h0' : fst κ a u = 0 := le_antisymm ((hκν a _).trans h0.le) zero_le'
rw [fst_apply' _ _ hu_meas] at h0'
refine (measure_mono_null ?_ h0').symm
intro p
simp only [mem_prod, mem_setOf_eq, and_imp]
exact fun h _ ↦ h
rw [div_eq_mul_inv, mul_assoc, ENNReal.inv_mul_cancel h0, mul_one]
exact measure_ne_top _ _
open scoped Function in -- required for scoped `on` notation | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.MapComap",
"Mathlib.Probability.Martingale.Convergence",
"Mathlib.Probability.Process.PartitionFiltration"
] | Mathlib/Probability/Kernel/Disintegration/Density.lean | setIntegral_densityProcess_of_mem | null |
setIntegral_densityProcess (hκν : fst κ ≤ ν) [IsFiniteKernel ν]
(n : ℕ) (a : α) {s : Set β} (hs : MeasurableSet s) {A : Set γ}
(hA : MeasurableSet[countableFiltration γ n] A) :
∫ x in A, densityProcess κ ν n a x s ∂(ν a) = (κ a).real (A ×ˢ s) := by
have : IsFiniteKernel κ := isFiniteKernel_of_isFiniteKernel_fst (h := isFiniteKernel_of_le hκν)
obtain ⟨S, hS_subset, rfl⟩ := (measurableSet_generateFrom_countablePartition_iff _ _).mp hA
simp_rw [sUnion_eq_iUnion]
have h_disj : Pairwise (Disjoint on fun i : S ↦ (i : Set γ)) := by
intro u v huv
simp only [Function.onFun]
refine disjoint_countablePartition (hS_subset (by simp)) (hS_subset (by simp)) ?_
rwa [ne_eq, ← Subtype.ext_iff]
rw [integral_iUnion, iUnion_prod_const, measureReal_def, measure_iUnion,
ENNReal.tsum_toReal_eq (fun _ ↦ measure_ne_top _ _)]
· congr with u
rw [setIntegral_densityProcess_of_mem hκν _ _ hs (hS_subset (by simp))]
rfl
· intro u v huv
simp only [Finset.coe_sort_coe, Set.disjoint_prod, disjoint_self, bot_eq_empty]
exact Or.inl (h_disj huv)
· exact fun _ ↦ (measurableSet_countablePartition n (hS_subset (by simp))).prod hs
· exact fun _ ↦ measurableSet_countablePartition n (hS_subset (by simp))
· exact h_disj
· exact (integrable_densityProcess hκν _ _ hs).integrableOn | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.MapComap",
"Mathlib.Probability.Martingale.Convergence",
"Mathlib.Probability.Process.PartitionFiltration"
] | Mathlib/Probability/Kernel/Disintegration/Density.lean | setIntegral_densityProcess | null |
integral_densityProcess (hκν : fst κ ≤ ν) [IsFiniteKernel ν]
(n : ℕ) (a : α) {s : Set β} (hs : MeasurableSet s) :
∫ x, densityProcess κ ν n a x s ∂(ν a) = (κ a).real (univ ×ˢ s) := by
rw [← setIntegral_univ, setIntegral_densityProcess hκν _ _ hs MeasurableSet.univ] | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.MapComap",
"Mathlib.Probability.Martingale.Convergence",
"Mathlib.Probability.Process.PartitionFiltration"
] | Mathlib/Probability/Kernel/Disintegration/Density.lean | integral_densityProcess | null |
setIntegral_densityProcess_of_le (hκν : fst κ ≤ ν)
[IsFiniteKernel ν] {n m : ℕ} (hnm : n ≤ m) (a : α) {s : Set β} (hs : MeasurableSet s)
{A : Set γ} (hA : MeasurableSet[countableFiltration γ n] A) :
∫ x in A, densityProcess κ ν m a x s ∂(ν a) = (κ a).real (A ×ˢ s) :=
setIntegral_densityProcess hκν m a hs ((countableFiltration γ).mono hnm A hA) | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.MapComap",
"Mathlib.Probability.Martingale.Convergence",
"Mathlib.Probability.Process.PartitionFiltration"
] | Mathlib/Probability/Kernel/Disintegration/Density.lean | setIntegral_densityProcess_of_le | null |
condExp_densityProcess (hκν : fst κ ≤ ν) [IsFiniteKernel ν]
{i j : ℕ} (hij : i ≤ j) (a : α) {s : Set β} (hs : MeasurableSet s) :
(ν a)[fun x ↦ densityProcess κ ν j a x s | countableFiltration γ i]
=ᵐ[ν a] fun x ↦ densityProcess κ ν i a x s := by
refine (ae_eq_condExp_of_forall_setIntegral_eq ?_ ?_ ?_ ?_ ?_).symm
· exact integrable_densityProcess hκν j a hs
· exact fun _ _ _ ↦ (integrable_densityProcess hκν _ _ hs).integrableOn
· intro x hx _
rw [setIntegral_densityProcess hκν i a hs hx,
setIntegral_densityProcess_of_le hκν hij a hs hx]
· exact StronglyMeasurable.aestronglyMeasurable
(stronglyMeasurable_countableFiltration_densityProcess κ ν i a hs) | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.MapComap",
"Mathlib.Probability.Martingale.Convergence",
"Mathlib.Probability.Process.PartitionFiltration"
] | Mathlib/Probability/Kernel/Disintegration/Density.lean | condExp_densityProcess | null |
martingale_densityProcess (hκν : fst κ ≤ ν) [IsFiniteKernel ν]
(a : α) {s : Set β} (hs : MeasurableSet s) :
Martingale (fun n x ↦ densityProcess κ ν n a x s) (countableFiltration γ) (ν a) :=
⟨adapted_densityProcess κ ν a hs, fun _ _ h ↦ condExp_densityProcess hκν h a hs⟩ | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.MapComap",
"Mathlib.Probability.Martingale.Convergence",
"Mathlib.Probability.Process.PartitionFiltration"
] | Mathlib/Probability/Kernel/Disintegration/Density.lean | martingale_densityProcess | null |
densityProcess_mono_set (hκν : fst κ ≤ ν) (n : ℕ) (a : α) (x : γ)
{s s' : Set β} (h : s ⊆ s') :
densityProcess κ ν n a x s ≤ densityProcess κ ν n a x s' := by
unfold densityProcess
obtain h₀ | h₀ := eq_or_ne (ν a (countablePartitionSet n x)) 0
· simp [h₀]
· gcongr
simp only [ne_eq, ENNReal.div_eq_top, h₀, and_false, false_or, not_and, not_not]
exact eq_top_mono (meas_countablePartitionSet_le_of_fst_le hκν n a x s') | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.MapComap",
"Mathlib.Probability.Martingale.Convergence",
"Mathlib.Probability.Process.PartitionFiltration"
] | Mathlib/Probability/Kernel/Disintegration/Density.lean | densityProcess_mono_set | null |
densityProcess_mono_kernel_left {κ' : Kernel α (γ × β)} (hκκ' : κ ≤ κ')
(hκ'ν : fst κ' ≤ ν) (n : ℕ) (a : α) (x : γ) (s : Set β) :
densityProcess κ ν n a x s ≤ densityProcess κ' ν n a x s := by
unfold densityProcess
by_cases h0 : ν a (countablePartitionSet n x) = 0
· rw [h0, ENNReal.toReal_div, ENNReal.toReal_div]
simp
have h_le : κ' a (countablePartitionSet n x ×ˢ s) ≤ ν a (countablePartitionSet n x) :=
meas_countablePartitionSet_le_of_fst_le hκ'ν n a x s
gcongr
· simp only [ne_eq, ENNReal.div_eq_top, h0, and_false, false_or, not_and, not_not]
exact fun h_top ↦ eq_top_mono h_le h_top
· apply hκκ' | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.MapComap",
"Mathlib.Probability.Martingale.Convergence",
"Mathlib.Probability.Process.PartitionFiltration"
] | Mathlib/Probability/Kernel/Disintegration/Density.lean | densityProcess_mono_kernel_left | null |
densityProcess_antitone_kernel_right {ν' : Kernel α γ}
(hνν' : ν ≤ ν') (hκν : fst κ ≤ ν) (n : ℕ) (a : α) (x : γ) (s : Set β) :
densityProcess κ ν' n a x s ≤ densityProcess κ ν n a x s := by
unfold densityProcess
have h_le : κ a (countablePartitionSet n x ×ˢ s) ≤ ν a (countablePartitionSet n x) :=
meas_countablePartitionSet_le_of_fst_le hκν n a x s
by_cases h0 : ν a (countablePartitionSet n x) = 0
· simp [le_antisymm (h_le.trans h0.le) zero_le', h0]
gcongr
· simp only [ne_eq, ENNReal.div_eq_top, h0, and_false, false_or, not_and, not_not]
exact fun h_top ↦ eq_top_mono h_le h_top
· apply hνν'
@[simp] | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.MapComap",
"Mathlib.Probability.Martingale.Convergence",
"Mathlib.Probability.Process.PartitionFiltration"
] | Mathlib/Probability/Kernel/Disintegration/Density.lean | densityProcess_antitone_kernel_right | null |
densityProcess_empty (κ : Kernel α (γ × β)) (ν : Kernel α γ) (n : ℕ) (a : α) (x : γ) :
densityProcess κ ν n a x ∅ = 0 := by
simp [densityProcess] | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.MapComap",
"Mathlib.Probability.Martingale.Convergence",
"Mathlib.Probability.Process.PartitionFiltration"
] | Mathlib/Probability/Kernel/Disintegration/Density.lean | densityProcess_empty | null |
tendsto_densityProcess_atTop_empty_of_antitone (κ : Kernel α (γ × β)) (ν : Kernel α γ)
[IsFiniteKernel κ] (n : ℕ) (a : α) (x : γ)
(seq : ℕ → Set β) (hseq : Antitone seq) (hseq_iInter : ⋂ i, seq i = ∅)
(hseq_meas : ∀ m, MeasurableSet (seq m)) :
Tendsto (fun m ↦ densityProcess κ ν n a x (seq m)) atTop
(𝓝 (densityProcess κ ν n a x ∅)) := by
simp_rw [densityProcess]
by_cases h0 : ν a (countablePartitionSet n x) = 0
· simp_rw [h0, ENNReal.toReal_div]
simp
refine (ENNReal.tendsto_toReal ?_).comp ?_
· rw [ne_eq, ENNReal.div_eq_top]
push_neg
simp
refine ENNReal.Tendsto.div_const ?_ (.inr h0)
have : Tendsto (fun m ↦ κ a (countablePartitionSet n x ×ˢ seq m)) atTop
(𝓝 ((κ a) (⋂ n_1, countablePartitionSet n x ×ˢ seq n_1))) := by
apply tendsto_measure_iInter_atTop
· measurability
· exact fun _ _ h ↦ prod_mono_right <| hseq h
· exact ⟨0, measure_ne_top _ _⟩
simpa only [← prod_iInter, hseq_iInter] using this | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.MapComap",
"Mathlib.Probability.Martingale.Convergence",
"Mathlib.Probability.Process.PartitionFiltration"
] | Mathlib/Probability/Kernel/Disintegration/Density.lean | tendsto_densityProcess_atTop_empty_of_antitone | null |
tendsto_densityProcess_atTop_of_antitone (κ : Kernel α (γ × β)) (ν : Kernel α γ)
[IsFiniteKernel κ] (n : ℕ) (a : α) (x : γ)
(seq : ℕ → Set β) (hseq : Antitone seq) (hseq_iInter : ⋂ i, seq i = ∅)
(hseq_meas : ∀ m, MeasurableSet (seq m)) :
Tendsto (fun m ↦ densityProcess κ ν n a x (seq m)) atTop (𝓝 0) := by
rw [← densityProcess_empty κ ν n a x]
exact tendsto_densityProcess_atTop_empty_of_antitone κ ν n a x seq hseq hseq_iInter hseq_meas | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.MapComap",
"Mathlib.Probability.Martingale.Convergence",
"Mathlib.Probability.Process.PartitionFiltration"
] | Mathlib/Probability/Kernel/Disintegration/Density.lean | tendsto_densityProcess_atTop_of_antitone | null |
tendsto_densityProcess_limitProcess (hκν : fst κ ≤ ν)
[IsFiniteKernel ν] (a : α) {s : Set β} (hs : MeasurableSet s) :
∀ᵐ x ∂(ν a), Tendsto (fun n ↦ densityProcess κ ν n a x s) atTop
(𝓝 ((countableFiltration γ).limitProcess
(fun n x ↦ densityProcess κ ν n a x s) (ν a) x)) := by
refine Submartingale.ae_tendsto_limitProcess (martingale_densityProcess hκν a hs).submartingale
(R := (ν a univ).toNNReal) (fun n ↦ ?_)
refine (eLpNorm_densityProcess_le hκν n a s).trans_eq ?_
rw [ENNReal.coe_toNNReal]
exact measure_ne_top _ _ | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.MapComap",
"Mathlib.Probability.Martingale.Convergence",
"Mathlib.Probability.Process.PartitionFiltration"
] | Mathlib/Probability/Kernel/Disintegration/Density.lean | tendsto_densityProcess_limitProcess | null |
memL1_limitProcess_densityProcess (hκν : fst κ ≤ ν) [IsFiniteKernel ν]
(a : α) {s : Set β} (hs : MeasurableSet s) :
MemLp ((countableFiltration γ).limitProcess
(fun n x ↦ densityProcess κ ν n a x s) (ν a)) 1 (ν a) := by
refine Submartingale.memLp_limitProcess (martingale_densityProcess hκν a hs).submartingale
(R := (ν a univ).toNNReal) (fun n ↦ ?_)
refine (eLpNorm_densityProcess_le hκν n a s).trans_eq ?_
rw [ENNReal.coe_toNNReal]
exact measure_ne_top _ _ | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.MapComap",
"Mathlib.Probability.Martingale.Convergence",
"Mathlib.Probability.Process.PartitionFiltration"
] | Mathlib/Probability/Kernel/Disintegration/Density.lean | memL1_limitProcess_densityProcess | null |
tendsto_eLpNorm_one_densityProcess_limitProcess (hκν : fst κ ≤ ν) [IsFiniteKernel ν]
(a : α) {s : Set β} (hs : MeasurableSet s) :
Tendsto (fun n ↦ eLpNorm ((fun x ↦ densityProcess κ ν n a x s)
- (countableFiltration γ).limitProcess (fun n x ↦ densityProcess κ ν n a x s) (ν a))
1 (ν a)) atTop (𝓝 0) := by
refine Submartingale.tendsto_eLpNorm_one_limitProcess ?_ ?_
· exact (martingale_densityProcess hκν a hs).submartingale
· refine uniformIntegrable_of le_rfl ENNReal.one_ne_top ?_ ?_
· exact fun n ↦ (measurable_densityProcess_right κ ν n a hs).aestronglyMeasurable
· refine fun ε _ ↦ ⟨2, fun n ↦ le_of_eq_of_le ?_ (?_ : 0 ≤ ENNReal.ofReal ε)⟩
· suffices {x | 2 ≤ ‖densityProcess κ ν n a x s‖₊} = ∅ by simp [this]
ext x
simp only [mem_setOf_eq, mem_empty_iff_false, iff_false, not_le]
refine (?_ : _ ≤ (1 : ℝ≥0)).trans_lt one_lt_two
rw [Real.nnnorm_of_nonneg (densityProcess_nonneg _ _ _ _ _ _)]
exact mod_cast (densityProcess_le_one hκν _ _ _ _)
· simp | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.MapComap",
"Mathlib.Probability.Martingale.Convergence",
"Mathlib.Probability.Process.PartitionFiltration"
] | Mathlib/Probability/Kernel/Disintegration/Density.lean | tendsto_eLpNorm_one_densityProcess_limitProcess | null |
tendsto_eLpNorm_one_restrict_densityProcess_limitProcess [IsFiniteKernel ν]
(hκν : fst κ ≤ ν) (a : α) {s : Set β} (hs : MeasurableSet s) (A : Set γ) :
Tendsto (fun n ↦ eLpNorm ((fun x ↦ densityProcess κ ν n a x s)
- (countableFiltration γ).limitProcess (fun n x ↦ densityProcess κ ν n a x s) (ν a))
1 ((ν a).restrict A)) atTop (𝓝 0) :=
tendsto_of_tendsto_of_tendsto_of_le_of_le tendsto_const_nhds
(tendsto_eLpNorm_one_densityProcess_limitProcess hκν a hs) (fun _ ↦ zero_le')
(fun _ ↦ eLpNorm_restrict_le _ _ _ _) | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.MapComap",
"Mathlib.Probability.Martingale.Convergence",
"Mathlib.Probability.Process.PartitionFiltration"
] | Mathlib/Probability/Kernel/Disintegration/Density.lean | tendsto_eLpNorm_one_restrict_densityProcess_limitProcess | null |
noncomputable
density (κ : Kernel α (γ × β)) (ν : Kernel α γ) (a : α) (x : γ) (s : Set β) : ℝ :=
limsup (fun n ↦ densityProcess κ ν n a x s) atTop | def | Probability | [
"Mathlib.Probability.Kernel.Composition.MapComap",
"Mathlib.Probability.Martingale.Convergence",
"Mathlib.Probability.Process.PartitionFiltration"
] | Mathlib/Probability/Kernel/Disintegration/Density.lean | density | Density of the kernel `κ` with respect to `ν`. This is a function `α → γ → Set β → ℝ` which
is measurable on `α × γ` for all measurable sets `s : Set β` and satisfies that
`∫ x in A, density κ ν a x s ∂(ν a) = (κ a).real (A ×ˢ s)` for all measurable `A : Set γ`. |
density_ae_eq_limitProcess (hκν : fst κ ≤ ν) [IsFiniteKernel ν]
(a : α) {s : Set β} (hs : MeasurableSet s) :
(fun x ↦ density κ ν a x s)
=ᵐ[ν a] (countableFiltration γ).limitProcess
(fun n x ↦ densityProcess κ ν n a x s) (ν a) := by
filter_upwards [tendsto_densityProcess_limitProcess hκν a hs] with t ht using ht.limsup_eq | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.MapComap",
"Mathlib.Probability.Martingale.Convergence",
"Mathlib.Probability.Process.PartitionFiltration"
] | Mathlib/Probability/Kernel/Disintegration/Density.lean | density_ae_eq_limitProcess | null |
tendsto_m_density (hκν : fst κ ≤ ν) (a : α) [IsFiniteKernel ν]
{s : Set β} (hs : MeasurableSet s) :
∀ᵐ x ∂(ν a),
Tendsto (fun n ↦ densityProcess κ ν n a x s) atTop (𝓝 (density κ ν a x s)) := by
filter_upwards [tendsto_densityProcess_limitProcess hκν a hs, density_ae_eq_limitProcess hκν a hs]
with t h1 h2 using h2 ▸ h1 | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.MapComap",
"Mathlib.Probability.Martingale.Convergence",
"Mathlib.Probability.Process.PartitionFiltration"
] | Mathlib/Probability/Kernel/Disintegration/Density.lean | tendsto_m_density | null |
measurable_density (κ : Kernel α (γ × β)) (ν : Kernel α γ)
{s : Set β} (hs : MeasurableSet s) :
Measurable (fun (p : α × γ) ↦ density κ ν p.1 p.2 s) :=
.limsup (fun n ↦ measurable_densityProcess κ ν n hs) | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.MapComap",
"Mathlib.Probability.Martingale.Convergence",
"Mathlib.Probability.Process.PartitionFiltration"
] | Mathlib/Probability/Kernel/Disintegration/Density.lean | measurable_density | null |
measurable_density_left (κ : Kernel α (γ × β)) (ν : Kernel α γ) (x : γ)
{s : Set β} (hs : MeasurableSet s) :
Measurable (fun a ↦ density κ ν a x s) := by
change Measurable ((fun (p : α × γ) ↦ density κ ν p.1 p.2 s) ∘ (fun a ↦ (a, x)))
exact (measurable_density κ ν hs).comp measurable_prodMk_right | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.MapComap",
"Mathlib.Probability.Martingale.Convergence",
"Mathlib.Probability.Process.PartitionFiltration"
] | Mathlib/Probability/Kernel/Disintegration/Density.lean | measurable_density_left | null |
measurable_density_right (κ : Kernel α (γ × β)) (ν : Kernel α γ)
{s : Set β} (hs : MeasurableSet s) (a : α) :
Measurable (fun x ↦ density κ ν a x s) := by
change Measurable ((fun (p : α × γ) ↦ density κ ν p.1 p.2 s) ∘ (fun x ↦ (a, x)))
exact (measurable_density κ ν hs).comp measurable_prodMk_left | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.MapComap",
"Mathlib.Probability.Martingale.Convergence",
"Mathlib.Probability.Process.PartitionFiltration"
] | Mathlib/Probability/Kernel/Disintegration/Density.lean | measurable_density_right | null |
density_mono_set (hκν : fst κ ≤ ν) (a : α) (x : γ) {s s' : Set β} (h : s ⊆ s') :
density κ ν a x s ≤ density κ ν a x s' := by
refine limsup_le_limsup ?_ ?_ ?_
· exact Eventually.of_forall (fun n ↦ densityProcess_mono_set hκν n a x h)
· exact isCoboundedUnder_le_of_le atTop (fun i ↦ densityProcess_nonneg _ _ _ _ _ _)
· exact isBoundedUnder_of ⟨1, fun n ↦ densityProcess_le_one hκν _ _ _ _⟩ | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.MapComap",
"Mathlib.Probability.Martingale.Convergence",
"Mathlib.Probability.Process.PartitionFiltration"
] | Mathlib/Probability/Kernel/Disintegration/Density.lean | density_mono_set | null |
density_nonneg (hκν : fst κ ≤ ν) (a : α) (x : γ) (s : Set β) :
0 ≤ density κ ν a x s := by
refine le_limsup_of_frequently_le ?_ ?_
· exact Frequently.of_forall (fun n ↦ densityProcess_nonneg _ _ _ _ _ _)
· exact isBoundedUnder_of ⟨1, fun n ↦ densityProcess_le_one hκν _ _ _ _⟩ | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.MapComap",
"Mathlib.Probability.Martingale.Convergence",
"Mathlib.Probability.Process.PartitionFiltration"
] | Mathlib/Probability/Kernel/Disintegration/Density.lean | density_nonneg | null |
density_le_one (hκν : fst κ ≤ ν) (a : α) (x : γ) (s : Set β) :
density κ ν a x s ≤ 1 := by
refine limsup_le_of_le ?_ ?_
· exact isCoboundedUnder_le_of_le atTop (fun i ↦ densityProcess_nonneg _ _ _ _ _ _)
· exact Eventually.of_forall (fun n ↦ densityProcess_le_one hκν _ _ _ _) | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.MapComap",
"Mathlib.Probability.Martingale.Convergence",
"Mathlib.Probability.Process.PartitionFiltration"
] | Mathlib/Probability/Kernel/Disintegration/Density.lean | density_le_one | null |
eLpNorm_density_le (hκν : fst κ ≤ ν) (a : α) (s : Set β) :
eLpNorm (fun x ↦ density κ ν a x s) 1 (ν a) ≤ ν a univ := by
refine (eLpNorm_le_of_ae_bound (C := 1) (ae_of_all _ (fun t ↦ ?_))).trans ?_
· simp only [Real.norm_eq_abs, abs_of_nonneg (density_nonneg hκν a t s),
density_le_one hκν a t s]
· simp | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.MapComap",
"Mathlib.Probability.Martingale.Convergence",
"Mathlib.Probability.Process.PartitionFiltration"
] | Mathlib/Probability/Kernel/Disintegration/Density.lean | eLpNorm_density_le | null |
integrable_density (hκν : fst κ ≤ ν) [IsFiniteKernel ν]
(a : α) {s : Set β} (hs : MeasurableSet s) :
Integrable (fun x ↦ density κ ν a x s) (ν a) := by
rw [← memLp_one_iff_integrable]
refine ⟨Measurable.aestronglyMeasurable ?_, ?_⟩
· exact measurable_density_right κ ν hs a
· exact (eLpNorm_density_le hκν a s).trans_lt (measure_lt_top _ _) | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.MapComap",
"Mathlib.Probability.Martingale.Convergence",
"Mathlib.Probability.Process.PartitionFiltration"
] | Mathlib/Probability/Kernel/Disintegration/Density.lean | integrable_density | null |
tendsto_setIntegral_densityProcess (hκν : fst κ ≤ ν)
[IsFiniteKernel ν] (a : α) {s : Set β} (hs : MeasurableSet s) (A : Set γ) :
Tendsto (fun i ↦ ∫ x in A, densityProcess κ ν i a x s ∂(ν a)) atTop
(𝓝 (∫ x in A, density κ ν a x s ∂(ν a))) := by
refine tendsto_setIntegral_of_L1' (μ := ν a) (fun x ↦ density κ ν a x s)
(integrable_density hκν a hs) (F := fun i x ↦ densityProcess κ ν i a x s) (l := atTop)
(Eventually.of_forall (fun n ↦ integrable_densityProcess hκν _ _ hs)) ?_ A
refine (tendsto_congr fun n ↦ ?_).mp (tendsto_eLpNorm_one_densityProcess_limitProcess hκν a hs)
refine eLpNorm_congr_ae ?_
exact EventuallyEq.rfl.sub (density_ae_eq_limitProcess hκν a hs).symm | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.MapComap",
"Mathlib.Probability.Martingale.Convergence",
"Mathlib.Probability.Process.PartitionFiltration"
] | Mathlib/Probability/Kernel/Disintegration/Density.lean | tendsto_setIntegral_densityProcess | null |
setIntegral_density_of_measurableSet (hκν : fst κ ≤ ν)
[IsFiniteKernel ν] (n : ℕ) (a : α) {s : Set β} (hs : MeasurableSet s) {A : Set γ}
(hA : MeasurableSet[countableFiltration γ n] A) :
∫ x in A, density κ ν a x s ∂(ν a) = (κ a).real (A ×ˢ s) := by
suffices ∫ x in A, density κ ν a x s ∂(ν a) = ∫ x in A, densityProcess κ ν n a x s ∂(ν a) by
exact this ▸ setIntegral_densityProcess hκν _ _ hs hA
suffices ∫ x in A, density κ ν a x s ∂(ν a)
= limsup (fun i ↦ ∫ x in A, densityProcess κ ν i a x s ∂(ν a)) atTop by
rw [this, ← limsup_const (α := ℕ) (f := atTop) (∫ x in A, densityProcess κ ν n a x s ∂(ν a)),
limsup_congr]
simp only [eventually_atTop]
refine ⟨n, fun m hnm ↦ ?_⟩
rw [setIntegral_densityProcess_of_le hκν hnm _ hs hA,
setIntegral_densityProcess hκν _ _ hs hA]
have h := tendsto_setIntegral_densityProcess hκν a hs A
rw [h.limsup_eq] | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.MapComap",
"Mathlib.Probability.Martingale.Convergence",
"Mathlib.Probability.Process.PartitionFiltration"
] | Mathlib/Probability/Kernel/Disintegration/Density.lean | setIntegral_density_of_measurableSet | Auxiliary lemma for `setIntegral_density`. |
integral_density (hκν : fst κ ≤ ν) [IsFiniteKernel ν]
(a : α) {s : Set β} (hs : MeasurableSet s) :
∫ x, density κ ν a x s ∂(ν a) = (κ a).real (univ ×ˢ s) := by
rw [← setIntegral_univ, setIntegral_density_of_measurableSet hκν 0 a hs MeasurableSet.univ] | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.MapComap",
"Mathlib.Probability.Martingale.Convergence",
"Mathlib.Probability.Process.PartitionFiltration"
] | Mathlib/Probability/Kernel/Disintegration/Density.lean | integral_density | null |
setIntegral_density (hκν : fst κ ≤ ν) [IsFiniteKernel ν]
(a : α) {s : Set β} (hs : MeasurableSet s) {A : Set γ} (hA : MeasurableSet A) :
∫ x in A, density κ ν a x s ∂(ν a) = (κ a).real (A ×ˢ s) := by
have : IsFiniteKernel κ := isFiniteKernel_of_isFiniteKernel_fst (h := isFiniteKernel_of_le hκν)
have hgen : ‹MeasurableSpace γ› =
.generateFrom {s | ∃ n, MeasurableSet[countableFiltration γ n] s} := by
rw [setOf_exists, generateFrom_iUnion_measurableSet (countableFiltration γ),
iSup_countableFiltration]
have hpi : IsPiSystem {s | ∃ n, MeasurableSet[countableFiltration γ n] s} := by
rw [setOf_exists]
exact isPiSystem_iUnion_of_monotone _
(fun n ↦ @isPiSystem_measurableSet _ (countableFiltration γ n))
fun _ _ ↦ (countableFiltration γ).mono
induction A, hA using induction_on_inter hgen hpi with
| empty => simp
| basic s hs =>
rcases hs with ⟨n, hn⟩
exact setIntegral_density_of_measurableSet hκν n a hs hn
| compl A hA hA_eq =>
have h := integral_add_compl hA (integrable_density hκν a hs)
rw [hA_eq, integral_density hκν a hs] at h
have : Aᶜ ×ˢ s = univ ×ˢ s \ A ×ˢ s := by
rw [prod_diff_prod, compl_eq_univ_diff]
simp
rw [this, measureReal_def,
measure_diff (by intro; simp) (hA.prod hs).nullMeasurableSet (measure_ne_top (κ a) _),
ENNReal.toReal_sub_of_le (measure_mono (by intro x; simp)) (measure_ne_top _ _)]
rw [eq_tsub_iff_add_eq_of_le, add_comm]
· exact h
· gcongr <;> simp
| iUnion f hf_disj hf h_eq =>
rw [integral_iUnion hf hf_disj (integrable_density hκν _ hs).integrableOn]
simp_rw [h_eq, measureReal_def]
rw [← ENNReal.tsum_toReal_eq (fun _ ↦ measure_ne_top _ _)]
congr
rw [iUnion_prod_const, measure_iUnion]
· exact hf_disj.mono fun _ _ h ↦ h.set_prod_left _ _
· exact fun i ↦ (hf i).prod hs | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.MapComap",
"Mathlib.Probability.Martingale.Convergence",
"Mathlib.Probability.Process.PartitionFiltration"
] | Mathlib/Probability/Kernel/Disintegration/Density.lean | setIntegral_density | null |
setLIntegral_density (hκν : fst κ ≤ ν) [IsFiniteKernel ν]
(a : α) {s : Set β} (hs : MeasurableSet s) {A : Set γ} (hA : MeasurableSet A) :
∫⁻ x in A, ENNReal.ofReal (density κ ν a x s) ∂(ν a) = κ a (A ×ˢ s) := by
have : IsFiniteKernel κ := isFiniteKernel_of_isFiniteKernel_fst (h := isFiniteKernel_of_le hκν)
rw [← ofReal_integral_eq_lintegral_ofReal]
· rw [setIntegral_density hκν a hs hA, measureReal_def,
ENNReal.ofReal_toReal (measure_ne_top _ _)]
· exact (integrable_density hκν a hs).restrict
· exact ae_of_all _ (fun _ ↦ density_nonneg hκν _ _ _) | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.MapComap",
"Mathlib.Probability.Martingale.Convergence",
"Mathlib.Probability.Process.PartitionFiltration"
] | Mathlib/Probability/Kernel/Disintegration/Density.lean | setLIntegral_density | null |
lintegral_density (hκν : fst κ ≤ ν) [IsFiniteKernel ν]
(a : α) {s : Set β} (hs : MeasurableSet s) :
∫⁻ x, ENNReal.ofReal (density κ ν a x s) ∂(ν a) = κ a (univ ×ˢ s) := by
rw [← setLIntegral_univ]
exact setLIntegral_density hκν a hs MeasurableSet.univ | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.MapComap",
"Mathlib.Probability.Martingale.Convergence",
"Mathlib.Probability.Process.PartitionFiltration"
] | Mathlib/Probability/Kernel/Disintegration/Density.lean | lintegral_density | null |
tendsto_integral_density_of_monotone (hκν : fst κ ≤ ν) [IsFiniteKernel ν]
(a : α) (seq : ℕ → Set β) (hseq : Monotone seq) (hseq_iUnion : ⋃ i, seq i = univ)
(hseq_meas : ∀ m, MeasurableSet (seq m)) :
Tendsto (fun m ↦ ∫ x, density κ ν a x (seq m) ∂(ν a)) atTop (𝓝 ((κ a).real univ)) := by
have : IsFiniteKernel κ := isFiniteKernel_of_isFiniteKernel_fst (h := isFiniteKernel_of_le hκν)
simp_rw [integral_density hκν a (hseq_meas _)]
have h_cont := ENNReal.continuousOn_toReal.continuousAt (x := κ a univ) ?_
swap
· rw [mem_nhds_iff]
refine ⟨Iio (κ a univ + 1), fun x hx ↦ ne_top_of_lt (?_ : x < κ a univ + 1), isOpen_Iio, ?_⟩
· simpa using hx
· simp only [mem_Iio]
exact ENNReal.lt_add_right (measure_ne_top _ _) one_ne_zero
refine h_cont.tendsto.comp ?_
convert tendsto_measure_iUnion_atTop (monotone_const.set_prod hseq)
rw [← prod_iUnion, hseq_iUnion, univ_prod_univ] | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.MapComap",
"Mathlib.Probability.Martingale.Convergence",
"Mathlib.Probability.Process.PartitionFiltration"
] | Mathlib/Probability/Kernel/Disintegration/Density.lean | tendsto_integral_density_of_monotone | null |
tendsto_integral_density_of_antitone (hκν : fst κ ≤ ν) [IsFiniteKernel ν] (a : α)
(seq : ℕ → Set β) (hseq : Antitone seq) (hseq_iInter : ⋂ i, seq i = ∅)
(hseq_meas : ∀ m, MeasurableSet (seq m)) :
Tendsto (fun m ↦ ∫ x, density κ ν a x (seq m) ∂(ν a)) atTop (𝓝 0) := by
have : IsFiniteKernel κ := isFiniteKernel_of_isFiniteKernel_fst (h := isFiniteKernel_of_le hκν)
simp_rw [integral_density hκν a (hseq_meas _)]
rw [← ENNReal.toReal_zero]
have h_cont := ENNReal.continuousAt_toReal ENNReal.zero_ne_top
refine h_cont.tendsto.comp ?_
have h : Tendsto (fun m ↦ κ a (univ ×ˢ seq m)) atTop
(𝓝 ((κ a) (⋂ n, (fun m ↦ univ ×ˢ seq m) n))) := by
apply tendsto_measure_iInter_atTop
· measurability
· exact antitone_const.set_prod hseq
· exact ⟨0, measure_ne_top _ _⟩
simpa [← prod_iInter, hseq_iInter] using h | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.MapComap",
"Mathlib.Probability.Martingale.Convergence",
"Mathlib.Probability.Process.PartitionFiltration"
] | Mathlib/Probability/Kernel/Disintegration/Density.lean | tendsto_integral_density_of_antitone | null |
tendsto_density_atTop_ae_of_antitone (hκν : fst κ ≤ ν) [IsFiniteKernel ν] (a : α)
(seq : ℕ → Set β) (hseq : Antitone seq) (hseq_iInter : ⋂ i, seq i = ∅)
(hseq_meas : ∀ m, MeasurableSet (seq m)) :
∀ᵐ x ∂(ν a), Tendsto (fun m ↦ density κ ν a x (seq m)) atTop (𝓝 0) := by
refine tendsto_of_integral_tendsto_of_antitone ?_ (integrable_const _) ?_ ?_ ?_
· exact fun m ↦ integrable_density hκν _ (hseq_meas m)
· rw [integral_zero]
exact tendsto_integral_density_of_antitone hκν a seq hseq hseq_iInter hseq_meas
· exact ae_of_all _ (fun c n m hnm ↦ density_mono_set hκν a c (hseq hnm))
· exact ae_of_all _ (fun x m ↦ density_nonneg hκν a x (seq m)) | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.MapComap",
"Mathlib.Probability.Martingale.Convergence",
"Mathlib.Probability.Process.PartitionFiltration"
] | Mathlib/Probability/Kernel/Disintegration/Density.lean | tendsto_density_atTop_ae_of_antitone | null |
densityProcess_fst_univ [IsFiniteKernel κ] (n : ℕ) (a : α) (x : γ) :
densityProcess κ (fst κ) n a x univ
= if fst κ a (countablePartitionSet n x) = 0 then 0 else 1 := by
rw [densityProcess]
split_ifs with h
· simp only [h]
by_cases h' : κ a (countablePartitionSet n x ×ˢ univ) = 0
· simp [h']
· simp
· rw [fst_apply' _ _ (measurableSet_countablePartitionSet _ _)]
have : countablePartitionSet n x ×ˢ univ = {p : γ × β | p.1 ∈ countablePartitionSet n x} := by
ext x
simp
rw [this, ENNReal.div_self]
· simp
· rwa [fst_apply' _ _ (measurableSet_countablePartitionSet _ _)] at h
· exact measure_ne_top _ _ | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.MapComap",
"Mathlib.Probability.Martingale.Convergence",
"Mathlib.Probability.Process.PartitionFiltration"
] | Mathlib/Probability/Kernel/Disintegration/Density.lean | densityProcess_fst_univ | null |
densityProcess_fst_univ_ae (κ : Kernel α (γ × β)) [IsFiniteKernel κ] (n : ℕ) (a : α) :
∀ᵐ x ∂(fst κ a), densityProcess κ (fst κ) n a x univ = 1 := by
rw [ae_iff]
have : {x | ¬ densityProcess κ (fst κ) n a x univ = 1}
⊆ {x | fst κ a (countablePartitionSet n x) = 0} := by
intro x hx
simp only [mem_setOf_eq] at hx ⊢
rw [densityProcess_fst_univ] at hx
simpa using hx
refine measure_mono_null this ?_
have : {x | fst κ a (countablePartitionSet n x) = 0}
⊆ ⋃ (u) (_ : u ∈ countablePartition γ n) (_ : fst κ a u = 0), u := by
intro t ht
simp only [mem_setOf_eq, mem_iUnion, exists_prop] at ht ⊢
exact ⟨countablePartitionSet n t, countablePartitionSet_mem _ _, ht,
mem_countablePartitionSet _ _⟩
refine measure_mono_null this ?_
rw [measure_biUnion]
· simp
· exact (finite_countablePartition _ _).countable
· intro s hs t ht hst
simp only [disjoint_iUnion_right, disjoint_iUnion_left]
exact fun _ _ ↦ disjoint_countablePartition hs ht hst
· intro s hs
by_cases h : fst κ a s = 0
· simp [h, measurableSet_countablePartition n hs]
· simp [h] | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.MapComap",
"Mathlib.Probability.Martingale.Convergence",
"Mathlib.Probability.Process.PartitionFiltration"
] | Mathlib/Probability/Kernel/Disintegration/Density.lean | densityProcess_fst_univ_ae | null |
tendsto_densityProcess_fst_atTop_univ_of_monotone (κ : Kernel α (γ × β)) (n : ℕ) (a : α)
(x : γ) (seq : ℕ → Set β) (hseq : Monotone seq) (hseq_iUnion : ⋃ i, seq i = univ) :
Tendsto (fun m ↦ densityProcess κ (fst κ) n a x (seq m)) atTop
(𝓝 (densityProcess κ (fst κ) n a x univ)) := by
simp_rw [densityProcess]
refine (ENNReal.tendsto_toReal ?_).comp ?_
· rw [ne_eq, ENNReal.div_eq_top]
push_neg
simp_rw [fst_apply' _ _ (measurableSet_countablePartitionSet _ _)]
constructor
· refine fun h h0 ↦ h (measure_mono_null (fun x ↦ ?_) h0)
simp only [mem_prod, mem_setOf_eq, and_imp]
exact fun h _ ↦ h
· refine fun h_top ↦ eq_top_mono (measure_mono (fun x ↦ ?_)) h_top
simp only [mem_prod, mem_setOf_eq, and_imp]
exact fun h _ ↦ h
by_cases h0 : fst κ a (countablePartitionSet n x) = 0
· rw [fst_apply' _ _ (measurableSet_countablePartitionSet _ _)] at h0 ⊢
suffices ∀ m, κ a (countablePartitionSet n x ×ˢ seq m) = 0 by
simp only [this, h0, ENNReal.zero_div, tendsto_const_nhds_iff]
suffices κ a (countablePartitionSet n x ×ˢ univ) = 0 by
simp only [this, ENNReal.zero_div]
convert h0
ext x
simp only [mem_prod, mem_univ, and_true, mem_setOf_eq]
refine fun m ↦ measure_mono_null (fun x ↦ ?_) h0
simp only [mem_prod, mem_setOf_eq, and_imp]
exact fun h _ ↦ h
refine ENNReal.Tendsto.div_const ?_ ?_
· convert tendsto_measure_iUnion_atTop (monotone_const.set_prod hseq)
rw [← prod_iUnion, hseq_iUnion]
· exact Or.inr h0 | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.MapComap",
"Mathlib.Probability.Martingale.Convergence",
"Mathlib.Probability.Process.PartitionFiltration"
] | Mathlib/Probability/Kernel/Disintegration/Density.lean | tendsto_densityProcess_fst_atTop_univ_of_monotone | null |
tendsto_densityProcess_fst_atTop_ae_of_monotone (κ : Kernel α (γ × β)) [IsFiniteKernel κ]
(n : ℕ) (a : α) (seq : ℕ → Set β) (hseq : Monotone seq) (hseq_iUnion : ⋃ i, seq i = univ) :
∀ᵐ x ∂(fst κ a), Tendsto (fun m ↦ densityProcess κ (fst κ) n a x (seq m)) atTop (𝓝 1) := by
filter_upwards [densityProcess_fst_univ_ae κ n a] with x hx
rw [← hx]
exact tendsto_densityProcess_fst_atTop_univ_of_monotone κ n a x seq hseq hseq_iUnion | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.MapComap",
"Mathlib.Probability.Martingale.Convergence",
"Mathlib.Probability.Process.PartitionFiltration"
] | Mathlib/Probability/Kernel/Disintegration/Density.lean | tendsto_densityProcess_fst_atTop_ae_of_monotone | null |
density_fst_univ (κ : Kernel α (γ × β)) [IsFiniteKernel κ] (a : α) :
∀ᵐ x ∂(fst κ a), density κ (fst κ) a x univ = 1 := by
have h := fun n ↦ densityProcess_fst_univ_ae κ n a
rw [← ae_all_iff] at h
filter_upwards [h] with x hx
simp [density, hx] | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.MapComap",
"Mathlib.Probability.Martingale.Convergence",
"Mathlib.Probability.Process.PartitionFiltration"
] | Mathlib/Probability/Kernel/Disintegration/Density.lean | density_fst_univ | null |
tendsto_density_fst_atTop_ae_of_monotone [IsFiniteKernel κ]
(a : α) (seq : ℕ → Set β) (hseq : Monotone seq) (hseq_iUnion : ⋃ i, seq i = univ)
(hseq_meas : ∀ m, MeasurableSet (seq m)) :
∀ᵐ x ∂(fst κ a), Tendsto (fun m ↦ density κ (fst κ) a x (seq m)) atTop (𝓝 1) := by
refine tendsto_of_integral_tendsto_of_monotone ?_ (integrable_const _) ?_ ?_ ?_
· exact fun m ↦ integrable_density le_rfl _ (hseq_meas m)
· rw [MeasureTheory.integral_const, smul_eq_mul, mul_one]
convert tendsto_integral_density_of_monotone (κ := κ) le_rfl a seq hseq hseq_iUnion hseq_meas
simp only [measureReal_def]
rw [fst_apply' _ _ MeasurableSet.univ]
simp only [mem_univ, setOf_true]
· exact ae_of_all _ (fun c n m hnm ↦ density_mono_set le_rfl a c (hseq hnm))
· exact ae_of_all _ (fun x m ↦ density_le_one le_rfl a x (seq m)) | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.MapComap",
"Mathlib.Probability.Martingale.Convergence",
"Mathlib.Probability.Process.PartitionFiltration"
] | Mathlib/Probability/Kernel/Disintegration/Density.lean | tendsto_density_fst_atTop_ae_of_monotone | null |
lintegral_condKernel_mem (a : α) {s : Set (β × Ω)} (hs : MeasurableSet s) :
∫⁻ x, Kernel.condKernel κ (a, x) (Prod.mk x ⁻¹' s) ∂(Kernel.fst κ a) = κ a s := by
conv_rhs => rw [← κ.disintegrate κ.condKernel]
simp_rw [Kernel.compProd_apply hs] | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.IntegralCompProd",
"Mathlib.Probability.Kernel.Disintegration.StandardBorel"
] | Mathlib/Probability/Kernel/Disintegration/Integral.lean | lintegral_condKernel_mem | null |
setLIntegral_condKernel_eq_measure_prod (a : α) {s : Set β} (hs : MeasurableSet s)
{t : Set Ω} (ht : MeasurableSet t) :
∫⁻ b in s, Kernel.condKernel κ (a, b) t ∂(Kernel.fst κ a) = κ a (s ×ˢ t) := by
have : κ a (s ×ˢ t) = (Kernel.fst κ ⊗ₖ Kernel.condKernel κ) a (s ×ˢ t) := by
congr; exact (κ.disintegrate _).symm
rw [this, Kernel.compProd_apply (hs.prod ht)]
classical
have : ∀ b, Kernel.condKernel κ (a, b) {c | (b, c) ∈ s ×ˢ t}
= s.indicator (fun b ↦ Kernel.condKernel κ (a, b) t) b := by
intro b
by_cases hb : b ∈ s <;> simp [hb]
simp_rw [Set.preimage, this]
rw [lintegral_indicator hs] | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.IntegralCompProd",
"Mathlib.Probability.Kernel.Disintegration.StandardBorel"
] | Mathlib/Probability/Kernel/Disintegration/Integral.lean | setLIntegral_condKernel_eq_measure_prod | null |
lintegral_condKernel (hf : Measurable f) (a : α) :
∫⁻ b, ∫⁻ ω, f (b, ω) ∂(Kernel.condKernel κ (a, b)) ∂(Kernel.fst κ a) = ∫⁻ x, f x ∂(κ a) := by
conv_rhs => rw [← κ.disintegrate κ.condKernel]
rw [Kernel.lintegral_compProd _ _ _ hf] | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.IntegralCompProd",
"Mathlib.Probability.Kernel.Disintegration.StandardBorel"
] | Mathlib/Probability/Kernel/Disintegration/Integral.lean | lintegral_condKernel | null |
setLIntegral_condKernel (hf : Measurable f) (a : α) {s : Set β}
(hs : MeasurableSet s) {t : Set Ω} (ht : MeasurableSet t) :
∫⁻ b in s, ∫⁻ ω in t, f (b, ω) ∂(Kernel.condKernel κ (a, b)) ∂(Kernel.fst κ a)
= ∫⁻ x in s ×ˢ t, f x ∂(κ a) := by
conv_rhs => rw [← κ.disintegrate κ.condKernel]
rw [Kernel.setLIntegral_compProd _ _ _ hf hs ht] | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.IntegralCompProd",
"Mathlib.Probability.Kernel.Disintegration.StandardBorel"
] | Mathlib/Probability/Kernel/Disintegration/Integral.lean | setLIntegral_condKernel | null |
setLIntegral_condKernel_univ_right (hf : Measurable f) (a : α) {s : Set β}
(hs : MeasurableSet s) :
∫⁻ b in s, ∫⁻ ω, f (b, ω) ∂(Kernel.condKernel κ (a, b)) ∂(Kernel.fst κ a)
= ∫⁻ x in s ×ˢ Set.univ, f x ∂(κ a) := by
rw [← setLIntegral_condKernel hf a hs MeasurableSet.univ]; simp_rw [Measure.restrict_univ] | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.IntegralCompProd",
"Mathlib.Probability.Kernel.Disintegration.StandardBorel"
] | Mathlib/Probability/Kernel/Disintegration/Integral.lean | setLIntegral_condKernel_univ_right | null |
setLIntegral_condKernel_univ_left (hf : Measurable f) (a : α) {t : Set Ω}
(ht : MeasurableSet t) :
∫⁻ b, ∫⁻ ω in t, f (b, ω) ∂(Kernel.condKernel κ (a, b)) ∂(Kernel.fst κ a)
= ∫⁻ x in Set.univ ×ˢ t, f x ∂(κ a) := by
rw [← setLIntegral_condKernel hf a MeasurableSet.univ ht]; simp_rw [Measure.restrict_univ] | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.IntegralCompProd",
"Mathlib.Probability.Kernel.Disintegration.StandardBorel"
] | Mathlib/Probability/Kernel/Disintegration/Integral.lean | setLIntegral_condKernel_univ_left | null |
_root_.MeasureTheory.AEStronglyMeasurable.integral_kernel_condKernel (a : α)
(hf : AEStronglyMeasurable f (κ a)) :
AEStronglyMeasurable (fun x ↦ ∫ y, f (x, y) ∂(Kernel.condKernel κ (a, x)))
(Kernel.fst κ a) := by
rw [← κ.disintegrate κ.condKernel] at hf
exact AEStronglyMeasurable.integral_kernel_compProd hf | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.IntegralCompProd",
"Mathlib.Probability.Kernel.Disintegration.StandardBorel"
] | Mathlib/Probability/Kernel/Disintegration/Integral.lean | _root_.MeasureTheory.AEStronglyMeasurable.integral_kernel_condKernel | null |
integral_condKernel (a : α) (hf : Integrable f (κ a)) :
∫ b, ∫ ω, f (b, ω) ∂(Kernel.condKernel κ (a, b)) ∂(Kernel.fst κ a) = ∫ x, f x ∂(κ a) := by
conv_rhs => rw [← κ.disintegrate κ.condKernel]
rw [← κ.disintegrate κ.condKernel] at hf
rw [integral_compProd hf] | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.IntegralCompProd",
"Mathlib.Probability.Kernel.Disintegration.StandardBorel"
] | Mathlib/Probability/Kernel/Disintegration/Integral.lean | integral_condKernel | null |
setIntegral_condKernel (a : α) {s : Set β} (hs : MeasurableSet s)
{t : Set Ω} (ht : MeasurableSet t) (hf : IntegrableOn f (s ×ˢ t) (κ a)) :
∫ b in s, ∫ ω in t, f (b, ω) ∂(Kernel.condKernel κ (a, b)) ∂(Kernel.fst κ a)
= ∫ x in s ×ˢ t, f x ∂(κ a) := by
conv_rhs => rw [← κ.disintegrate κ.condKernel]
rw [← κ.disintegrate κ.condKernel] at hf
rw [setIntegral_compProd hs ht hf] | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.IntegralCompProd",
"Mathlib.Probability.Kernel.Disintegration.StandardBorel"
] | Mathlib/Probability/Kernel/Disintegration/Integral.lean | setIntegral_condKernel | null |
setIntegral_condKernel_univ_right (a : α) {s : Set β} (hs : MeasurableSet s)
(hf : IntegrableOn f (s ×ˢ Set.univ) (κ a)) :
∫ b in s, ∫ ω, f (b, ω) ∂(Kernel.condKernel κ (a, b)) ∂(Kernel.fst κ a)
= ∫ x in s ×ˢ Set.univ, f x ∂(κ a) := by
rw [← setIntegral_condKernel a hs MeasurableSet.univ hf]; simp_rw [Measure.restrict_univ] | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.IntegralCompProd",
"Mathlib.Probability.Kernel.Disintegration.StandardBorel"
] | Mathlib/Probability/Kernel/Disintegration/Integral.lean | setIntegral_condKernel_univ_right | null |
setIntegral_condKernel_univ_left (a : α) {t : Set Ω} (ht : MeasurableSet t)
(hf : IntegrableOn f (Set.univ ×ˢ t) (κ a)) :
∫ b, ∫ ω in t, f (b, ω) ∂(Kernel.condKernel κ (a, b)) ∂(Kernel.fst κ a)
= ∫ x in Set.univ ×ˢ t, f x ∂(κ a) := by
rw [← setIntegral_condKernel a MeasurableSet.univ ht hf]; simp_rw [Measure.restrict_univ] | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.IntegralCompProd",
"Mathlib.Probability.Kernel.Disintegration.StandardBorel"
] | Mathlib/Probability/Kernel/Disintegration/Integral.lean | setIntegral_condKernel_univ_left | null |
lintegral_condKernel_mem {s : Set (β × Ω)} (hs : MeasurableSet s) :
∫⁻ x, ρ.condKernel x {y | (x, y) ∈ s} ∂ρ.fst = ρ s := by
conv_rhs => rw [← ρ.disintegrate ρ.condKernel]
simp_rw [compProd_apply hs]
rfl | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.IntegralCompProd",
"Mathlib.Probability.Kernel.Disintegration.StandardBorel"
] | Mathlib/Probability/Kernel/Disintegration/Integral.lean | lintegral_condKernel_mem | null |
setLIntegral_condKernel_eq_measure_prod {s : Set β} (hs : MeasurableSet s) {t : Set Ω}
(ht : MeasurableSet t) :
∫⁻ b in s, ρ.condKernel b t ∂ρ.fst = ρ (s ×ˢ t) := by
have : ρ (s ×ˢ t) = (ρ.fst ⊗ₘ ρ.condKernel) (s ×ˢ t) := by
congr; exact (ρ.disintegrate _).symm
rw [this, compProd_apply (hs.prod ht)]
classical
have : ∀ b, ρ.condKernel b (Prod.mk b ⁻¹' s ×ˢ t)
= s.indicator (fun b ↦ ρ.condKernel b t) b := by
intro b
by_cases hb : b ∈ s <;> simp [hb]
simp_rw [this]
rw [lintegral_indicator hs] | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.IntegralCompProd",
"Mathlib.Probability.Kernel.Disintegration.StandardBorel"
] | Mathlib/Probability/Kernel/Disintegration/Integral.lean | setLIntegral_condKernel_eq_measure_prod | null |
lintegral_condKernel (hf : Measurable f) :
∫⁻ b, ∫⁻ ω, f (b, ω) ∂(ρ.condKernel b) ∂ρ.fst = ∫⁻ x, f x ∂ρ := by
conv_rhs => rw [← ρ.disintegrate ρ.condKernel]
rw [lintegral_compProd hf] | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.IntegralCompProd",
"Mathlib.Probability.Kernel.Disintegration.StandardBorel"
] | Mathlib/Probability/Kernel/Disintegration/Integral.lean | lintegral_condKernel | null |
setLIntegral_condKernel (hf : Measurable f) {s : Set β}
(hs : MeasurableSet s) {t : Set Ω} (ht : MeasurableSet t) :
∫⁻ b in s, ∫⁻ ω in t, f (b, ω) ∂(ρ.condKernel b) ∂ρ.fst
= ∫⁻ x in s ×ˢ t, f x ∂ρ := by
conv_rhs => rw [← ρ.disintegrate ρ.condKernel]
rw [setLIntegral_compProd hf hs ht] | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.IntegralCompProd",
"Mathlib.Probability.Kernel.Disintegration.StandardBorel"
] | Mathlib/Probability/Kernel/Disintegration/Integral.lean | setLIntegral_condKernel | null |
setLIntegral_condKernel_univ_right (hf : Measurable f) {s : Set β}
(hs : MeasurableSet s) :
∫⁻ b in s, ∫⁻ ω, f (b, ω) ∂(ρ.condKernel b) ∂ρ.fst
= ∫⁻ x in s ×ˢ Set.univ, f x ∂ρ := by
rw [← setLIntegral_condKernel hf hs MeasurableSet.univ]; simp_rw [Measure.restrict_univ] | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.IntegralCompProd",
"Mathlib.Probability.Kernel.Disintegration.StandardBorel"
] | Mathlib/Probability/Kernel/Disintegration/Integral.lean | setLIntegral_condKernel_univ_right | null |
setLIntegral_condKernel_univ_left (hf : Measurable f) {t : Set Ω}
(ht : MeasurableSet t) :
∫⁻ b, ∫⁻ ω in t, f (b, ω) ∂(ρ.condKernel b) ∂ρ.fst
= ∫⁻ x in Set.univ ×ˢ t, f x ∂ρ := by
rw [← setLIntegral_condKernel hf MeasurableSet.univ ht]; simp_rw [Measure.restrict_univ] | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.IntegralCompProd",
"Mathlib.Probability.Kernel.Disintegration.StandardBorel"
] | Mathlib/Probability/Kernel/Disintegration/Integral.lean | setLIntegral_condKernel_univ_left | null |
_root_.MeasureTheory.AEStronglyMeasurable.integral_condKernel
(hf : AEStronglyMeasurable f ρ) :
AEStronglyMeasurable (fun x ↦ ∫ y, f (x, y) ∂ρ.condKernel x) ρ.fst := by
rw [← ρ.disintegrate ρ.condKernel] at hf
exact AEStronglyMeasurable.integral_kernel_compProd hf | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.IntegralCompProd",
"Mathlib.Probability.Kernel.Disintegration.StandardBorel"
] | Mathlib/Probability/Kernel/Disintegration/Integral.lean | _root_.MeasureTheory.AEStronglyMeasurable.integral_condKernel | null |
integral_condKernel (hf : Integrable f ρ) :
∫ b, ∫ ω, f (b, ω) ∂(ρ.condKernel b) ∂ρ.fst = ∫ x, f x ∂ρ := by
conv_rhs => rw [← ρ.disintegrate ρ.condKernel]
rw [← ρ.disintegrate ρ.condKernel] at hf
rw [integral_compProd hf] | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.IntegralCompProd",
"Mathlib.Probability.Kernel.Disintegration.StandardBorel"
] | Mathlib/Probability/Kernel/Disintegration/Integral.lean | integral_condKernel | null |
setIntegral_condKernel {s : Set β} (hs : MeasurableSet s)
{t : Set Ω} (ht : MeasurableSet t) (hf : IntegrableOn f (s ×ˢ t) ρ) :
∫ b in s, ∫ ω in t, f (b, ω) ∂(ρ.condKernel b) ∂ρ.fst = ∫ x in s ×ˢ t, f x ∂ρ := by
conv_rhs => rw [← ρ.disintegrate ρ.condKernel]
rw [← ρ.disintegrate ρ.condKernel] at hf
rw [setIntegral_compProd hs ht hf] | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.IntegralCompProd",
"Mathlib.Probability.Kernel.Disintegration.StandardBorel"
] | Mathlib/Probability/Kernel/Disintegration/Integral.lean | setIntegral_condKernel | null |
setIntegral_condKernel_univ_right {s : Set β} (hs : MeasurableSet s)
(hf : IntegrableOn f (s ×ˢ Set.univ) ρ) :
∫ b in s, ∫ ω, f (b, ω) ∂(ρ.condKernel b) ∂ρ.fst = ∫ x in s ×ˢ Set.univ, f x ∂ρ := by
rw [← setIntegral_condKernel hs MeasurableSet.univ hf]; simp_rw [Measure.restrict_univ] | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.IntegralCompProd",
"Mathlib.Probability.Kernel.Disintegration.StandardBorel"
] | Mathlib/Probability/Kernel/Disintegration/Integral.lean | setIntegral_condKernel_univ_right | null |
setIntegral_condKernel_univ_left {t : Set Ω} (ht : MeasurableSet t)
(hf : IntegrableOn f (Set.univ ×ˢ t) ρ) :
∫ b, ∫ ω in t, f (b, ω) ∂(ρ.condKernel b) ∂ρ.fst = ∫ x in Set.univ ×ˢ t, f x ∂ρ := by
rw [← setIntegral_condKernel MeasurableSet.univ ht hf]; simp_rw [Measure.restrict_univ] | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.IntegralCompProd",
"Mathlib.Probability.Kernel.Disintegration.StandardBorel"
] | Mathlib/Probability/Kernel/Disintegration/Integral.lean | setIntegral_condKernel_univ_left | null |
AEStronglyMeasurable.ae_integrable_condKernel_iff {f : α × Ω → F}
(hf : AEStronglyMeasurable f ρ) :
(∀ᵐ a ∂ρ.fst, Integrable (fun ω ↦ f (a, ω)) (ρ.condKernel a)) ∧
Integrable (fun a ↦ ∫ ω, ‖f (a, ω)‖ ∂ρ.condKernel a) ρ.fst ↔ Integrable f ρ := by
rw [← ρ.disintegrate ρ.condKernel] at hf
conv_rhs => rw [← ρ.disintegrate ρ.condKernel]
rw [Measure.integrable_compProd_iff hf] | theorem | Probability | [
"Mathlib.Probability.Kernel.Composition.IntegralCompProd",
"Mathlib.Probability.Kernel.Disintegration.StandardBorel"
] | Mathlib/Probability/Kernel/Disintegration/Integral.lean | AEStronglyMeasurable.ae_integrable_condKernel_iff | null |
Integrable.condKernel_ae {f : α × Ω → F} (hf_int : Integrable f ρ) :
∀ᵐ a ∂ρ.fst, Integrable (fun ω ↦ f (a, ω)) (ρ.condKernel a) := by
have hf_ae : AEStronglyMeasurable f ρ := hf_int.1
rw [← hf_ae.ae_integrable_condKernel_iff] at hf_int
exact hf_int.1 | theorem | Probability | [
"Mathlib.Probability.Kernel.Composition.IntegralCompProd",
"Mathlib.Probability.Kernel.Disintegration.StandardBorel"
] | Mathlib/Probability/Kernel/Disintegration/Integral.lean | Integrable.condKernel_ae | null |
Integrable.integral_norm_condKernel {f : α × Ω → F} (hf_int : Integrable f ρ) :
Integrable (fun x ↦ ∫ y, ‖f (x, y)‖ ∂ρ.condKernel x) ρ.fst := by
have hf_ae : AEStronglyMeasurable f ρ := hf_int.1
rw [← hf_ae.ae_integrable_condKernel_iff] at hf_int
exact hf_int.2 | theorem | Probability | [
"Mathlib.Probability.Kernel.Composition.IntegralCompProd",
"Mathlib.Probability.Kernel.Disintegration.StandardBorel"
] | Mathlib/Probability/Kernel/Disintegration/Integral.lean | Integrable.integral_norm_condKernel | null |
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