statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
value | commit stringclasses 1
value |
|---|---|---|---|---|---|---|---|---|---|---|
coe_fn_sup {f g : filtration ι m} : ⇑(f ⊔ g) = f ⊔ g | rfl | lemma | measure_theory.filtration.coe_fn_sup | probability.process | src/probability/process/filtration.lean | [
"measure_theory.function.conditional_expectation.real"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_fn_inf {f g : filtration ι m} : ⇑(f ⊓ g) = f ⊓ g | rfl | lemma | measure_theory.filtration.coe_fn_inf | probability.process | src/probability/process/filtration.lean | [
"measure_theory.function.conditional_expectation.real"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Sup_def (s : set (filtration ι m)) (i : ι) :
Sup s i = Sup ((λ f : filtration ι m, f i) '' s) | rfl | lemma | measure_theory.filtration.Sup_def | probability.process | src/probability/process/filtration.lean | [
"measure_theory.function.conditional_expectation.real"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Inf_def (s : set (filtration ι m)) (i : ι) :
Inf s i = if set.nonempty s then Inf ((λ f : filtration ι m, f i) '' s) else m | rfl | lemma | measure_theory.filtration.Inf_def | probability.process | src/probability/process/filtration.lean | [
"measure_theory.function.conditional_expectation.real"
] | [
"set.nonempty"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
measurable_set_of_filtration [preorder ι] {f : filtration ι m} {s : set Ω} {i : ι}
(hs : measurable_set[f i] s) : measurable_set[m] s | f.le i s hs | lemma | measure_theory.measurable_set_of_filtration | probability.process | src/probability/process/filtration.lean | [
"measure_theory.function.conditional_expectation.real"
] | [
"measurable_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sigma_finite_filtration [preorder ι] (μ : measure Ω) (f : filtration ι m) : Prop | (sigma_finite : ∀ i : ι, sigma_finite (μ.trim (f.le i))) | class | measure_theory.sigma_finite_filtration | probability.process | src/probability/process/filtration.lean | [
"measure_theory.function.conditional_expectation.real"
] | [] | A measure is σ-finite with respect to filtration if it is σ-finite with respect
to all the sub-σ-algebra of the filtration. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
sigma_finite_of_sigma_finite_filtration [preorder ι] (μ : measure Ω) (f : filtration ι m)
[hf : sigma_finite_filtration μ f] (i : ι) :
sigma_finite (μ.trim (f.le i)) | by apply hf.sigma_finite | instance | measure_theory.sigma_finite_of_sigma_finite_filtration | probability.process | src/probability/process/filtration.lean | [
"measure_theory.function.conditional_expectation.real"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_finite_measure.sigma_finite_filtration [preorder ι] (μ : measure Ω) (f : filtration ι m)
[is_finite_measure μ] :
sigma_finite_filtration μ f | ⟨λ n, by apply_instance⟩ | instance | measure_theory.is_finite_measure.sigma_finite_filtration | probability.process | src/probability/process/filtration.lean | [
"measure_theory.function.conditional_expectation.real"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
integrable.uniform_integrable_condexp_filtration
[preorder ι] {μ : measure Ω} [is_finite_measure μ] {f : filtration ι m}
{g : Ω → ℝ} (hg : integrable g μ) :
uniform_integrable (λ i, μ[g | f i]) 1 μ | hg.uniform_integrable_condexp f.le | lemma | measure_theory.integrable.uniform_integrable_condexp_filtration | probability.process | src/probability/process/filtration.lean | [
"measure_theory.function.conditional_expectation.real"
] | [] | Given a integrable function `g`, the conditional expectations of `g` with respect to a
filtration is uniformly integrable. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
filtration_of_set {s : ι → set Ω} (hsm : ∀ i, measurable_set (s i)) : filtration ι m | { seq := λ i, measurable_space.generate_from {t | ∃ j ≤ i, s j = t},
mono' := λ n m hnm, measurable_space.generate_from_mono
(λ t ⟨k, hk₁, hk₂⟩, ⟨k, hk₁.trans hnm, hk₂⟩),
le' := λ n, measurable_space.generate_from_le (λ t ⟨k, hk₁, hk₂⟩, hk₂ ▸ hsm k) } | def | measure_theory.filtration_of_set | probability.process | src/probability/process/filtration.lean | [
"measure_theory.function.conditional_expectation.real"
] | [
"measurable_set",
"measurable_space.generate_from",
"measurable_space.generate_from_le",
"measurable_space.generate_from_mono"
] | Given a sequence of measurable sets `(sₙ)`, `filtration_of_set` is the smallest filtration
such that `sₙ` is measurable with respect to the `n`-the sub-σ-algebra in `filtration_of_set`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
measurable_set_filtration_of_set {s : ι → set Ω}
(hsm : ∀ i, measurable_set[m] (s i)) (i : ι) {j : ι} (hj : j ≤ i) :
measurable_set[filtration_of_set hsm i] (s j) | measurable_space.measurable_set_generate_from ⟨j, hj, rfl⟩ | lemma | measure_theory.measurable_set_filtration_of_set | probability.process | src/probability/process/filtration.lean | [
"measure_theory.function.conditional_expectation.real"
] | [
"measurable_set",
"measurable_space.measurable_set_generate_from"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
measurable_set_filtration_of_set' {s : ι → set Ω}
(hsm : ∀ n, measurable_set[m] (s n)) (i : ι) :
measurable_set[filtration_of_set hsm i] (s i) | measurable_set_filtration_of_set hsm i le_rfl | lemma | measure_theory.measurable_set_filtration_of_set' | probability.process | src/probability/process/filtration.lean | [
"measure_theory.function.conditional_expectation.real"
] | [
"le_rfl",
"measurable_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
natural (u : ι → Ω → β) (hum : ∀ i, strongly_measurable (u i)) : filtration ι m | { seq := λ i, ⨆ j ≤ i, measurable_space.comap (u j) mβ,
mono' := λ i j hij, bsupr_mono $ λ k, ge_trans hij,
le' := λ i,
begin
refine supr₂_le _,
rintros j hj s ⟨t, ht, rfl⟩,
exact (hum j).measurable ht,
end } | def | measure_theory.filtration.natural | probability.process | src/probability/process/filtration.lean | [
"measure_theory.function.conditional_expectation.real"
] | [
"bsupr_mono",
"measurable",
"measurable_space.comap",
"supr₂_le"
] | Given a sequence of functions, the natural filtration is the smallest sequence
of σ-algebras such that that sequence of functions is measurable with respect to
the filtration. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
filtration_of_set_eq_natural [mul_zero_one_class β] [nontrivial β]
{s : ι → set Ω} (hsm : ∀ i, measurable_set[m] (s i)) :
filtration_of_set hsm = natural (λ i, (s i).indicator (λ ω, 1 : Ω → β))
(λ i, strongly_measurable_one.indicator (hsm i)) | begin
simp only [natural, filtration_of_set, measurable_space_supr_eq],
ext1 i,
refine le_antisymm (generate_from_le _) (generate_from_le _),
{ rintro _ ⟨j, hij, rfl⟩,
refine measurable_set_generate_from ⟨j, measurable_set_generate_from ⟨hij, _⟩⟩,
rw comap_eq_generate_from,
refine measurable_set_gen... | lemma | measure_theory.filtration.filtration_of_set_eq_natural | probability.process | src/probability/process/filtration.lean | [
"measure_theory.function.conditional_expectation.real"
] | [
"measurable_set",
"measurable_set.compl",
"measurable_set.univ",
"measurable_space.comap",
"measurable_space.generate_from",
"mul_zero_one_class",
"nontrivial",
"set.mem_singleton_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
limit_process (f : ι → Ω → E) (ℱ : filtration ι m) (μ : measure Ω . volume_tac) | if h : ∃ g : Ω → E, strongly_measurable[⨆ n, ℱ n] g ∧
∀ᵐ ω ∂μ, tendsto (λ n, f n ω) at_top (𝓝 (g ω)) then classical.some h else 0 | def | measure_theory.filtration.limit_process | probability.process | src/probability/process/filtration.lean | [
"measure_theory.function.conditional_expectation.real"
] | [] | Given a process `f` and a filtration `ℱ`, if `f` converges to some `g` almost everywhere and
`g` is `⨆ n, ℱ n`-measurable, then `limit_process f ℱ μ` chooses said `g`, else it returns 0.
This definition is used to phrase the a.e. martingale convergence theorem
`submartingale.ae_tendsto_limit_process` where an L¹-bound... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
strongly_measurable_limit_process :
strongly_measurable[⨆ n, ℱ n] (limit_process f ℱ μ) | begin
rw limit_process,
split_ifs with h h,
exacts [(classical.some_spec h).1, strongly_measurable_zero]
end | lemma | measure_theory.filtration.strongly_measurable_limit_process | probability.process | src/probability/process/filtration.lean | [
"measure_theory.function.conditional_expectation.real"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strongly_measurable_limit_process' :
strongly_measurable[m] (limit_process f ℱ μ) | strongly_measurable_limit_process.mono (Sup_le (λ m ⟨n, hn⟩, hn ▸ ℱ.le _)) | lemma | measure_theory.filtration.strongly_measurable_limit_process' | probability.process | src/probability/process/filtration.lean | [
"measure_theory.function.conditional_expectation.real"
] | [
"Sup_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_ℒp_limit_process_of_snorm_bdd {R : ℝ≥0} {p : ℝ≥0∞}
{F : Type*} [normed_add_comm_group F] {ℱ : filtration ℕ m} {f : ℕ → Ω → F}
(hfm : ∀ n, ae_strongly_measurable (f n) μ) (hbdd : ∀ n, snorm (f n) p μ ≤ R) :
mem_ℒp (limit_process f ℱ μ) p μ | begin
rw limit_process,
split_ifs with h,
{ refine ⟨strongly_measurable.ae_strongly_measurable
((classical.some_spec h).1.mono (Sup_le (λ m ⟨n, hn⟩, hn ▸ ℱ.le _))),
lt_of_le_of_lt (Lp.snorm_lim_le_liminf_snorm hfm _ (classical.some_spec h).2)
(lt_of_le_of_lt _ (ennreal.coe_lt_top : ↑R < ∞))⟩,
... | lemma | measure_theory.filtration.mem_ℒp_limit_process_of_snorm_bdd | probability.process | src/probability/process/filtration.lean | [
"measure_theory.function.conditional_expectation.real"
] | [
"Sup_le",
"ennreal.coe_lt_top",
"le_rfl",
"normed_add_comm_group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hitting [preorder ι] [has_Inf ι] (u : ι → Ω → β) (s : set β) (n m : ι) : Ω → ι | λ x, if ∃ j ∈ set.Icc n m, u j x ∈ s then Inf (set.Icc n m ∩ {i : ι | u i x ∈ s}) else m | def | measure_theory.hitting | probability.process | src/probability/process/hitting_time.lean | [
"probability.process.stopping"
] | [
"has_Inf",
"set.Icc"
] | Hitting time: given a stochastic process `u` and a set `s`, `hitting u s n m` is the first time
`u` is in `s` after time `n` and before time `m` (if `u` does not hit `s` after time `n` and
before `m` then the hitting time is simply `m`).
The hitting time is a stopping time if the process is adapted and discrete. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
hitting_of_lt {m : ι} (h : m < n) : hitting u s n m ω = m | begin
simp_rw [hitting],
have h_not : ¬ ∃ (j : ι) (H : j ∈ set.Icc n m), u j ω ∈ s,
{ push_neg,
intro j,
rw set.Icc_eq_empty_of_lt h,
simp only [set.mem_empty_iff_false, is_empty.forall_iff], },
simp only [h_not, if_false],
end | lemma | measure_theory.hitting_of_lt | probability.process | src/probability/process/hitting_time.lean | [
"probability.process.stopping"
] | [
"is_empty.forall_iff",
"set.Icc",
"set.Icc_eq_empty_of_lt",
"set.mem_empty_iff_false"
] | This lemma is strictly weaker than `hitting_of_le`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
hitting_le {m : ι} (ω : Ω) : hitting u s n m ω ≤ m | begin
cases le_or_lt n m with h_le h_lt,
{ simp only [hitting],
split_ifs,
{ obtain ⟨j, hj₁, hj₂⟩ := h,
exact (cInf_le (bdd_below.inter_of_left bdd_below_Icc) (set.mem_inter hj₁ hj₂)).trans hj₁.2 },
{ exact le_rfl }, },
{ rw hitting_of_lt h_lt, },
end | lemma | measure_theory.hitting_le | probability.process | src/probability/process/hitting_time.lean | [
"probability.process.stopping"
] | [
"bdd_below.inter_of_left",
"bdd_below_Icc",
"cInf_le",
"le_rfl",
"set.mem_inter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
not_mem_of_lt_hitting {m k : ι}
(hk₁ : k < hitting u s n m ω) (hk₂ : n ≤ k) :
u k ω ∉ s | begin
classical,
intro h,
have hexists : ∃ j ∈ set.Icc n m, u j ω ∈ s,
refine ⟨k, ⟨hk₂, le_trans hk₁.le $ hitting_le _⟩, h⟩,
refine not_le.2 hk₁ _,
simp_rw [hitting, if_pos hexists],
exact cInf_le bdd_below_Icc.inter_of_left ⟨⟨hk₂, le_trans hk₁.le $ hitting_le _⟩, h⟩,
end | lemma | measure_theory.not_mem_of_lt_hitting | probability.process | src/probability/process/hitting_time.lean | [
"probability.process.stopping"
] | [
"cInf_le",
"set.Icc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hitting_eq_end_iff {m : ι} :
hitting u s n m ω = m ↔ (∃ j ∈ set.Icc n m, u j ω ∈ s) →
Inf (set.Icc n m ∩ {i : ι | u i ω ∈ s}) = m | by rw [hitting, ite_eq_right_iff] | lemma | measure_theory.hitting_eq_end_iff | probability.process | src/probability/process/hitting_time.lean | [
"probability.process.stopping"
] | [
"ite_eq_right_iff",
"set.Icc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hitting_of_le {m : ι} (hmn : m ≤ n) :
hitting u s n m ω = m | begin
obtain (rfl | h) := le_iff_eq_or_lt.1 hmn,
{ simp only [hitting, set.Icc_self, ite_eq_right_iff, set.mem_Icc, exists_prop,
forall_exists_index, and_imp],
intros i hi₁ hi₂ hi,
rw [set.inter_eq_left_iff_subset.2, cInf_singleton],
exact set.singleton_subset_iff.2 (le_antisymm hi₂ hi₁ ▸ hi) },
... | lemma | measure_theory.hitting_of_le | probability.process | src/probability/process/hitting_time.lean | [
"probability.process.stopping"
] | [
"and_imp",
"cInf_singleton",
"exists_prop",
"forall_exists_index",
"ite_eq_right_iff",
"set.Icc_self",
"set.mem_Icc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_hitting {m : ι} (hnm : n ≤ m) (ω : Ω) : n ≤ hitting u s n m ω | begin
simp only [hitting],
split_ifs,
{ refine le_cInf _ (λ b hb, _),
{ obtain ⟨k, hk_Icc, hk_s⟩ := h,
exact ⟨k, hk_Icc, hk_s⟩, },
{ rw set.mem_inter_iff at hb,
exact hb.1.1, }, },
{ exact hnm },
end | lemma | measure_theory.le_hitting | probability.process | src/probability/process/hitting_time.lean | [
"probability.process.stopping"
] | [
"le_cInf",
"set.mem_inter_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_hitting_of_exists {m : ι} (h_exists : ∃ j ∈ set.Icc n m, u j ω ∈ s) :
n ≤ hitting u s n m ω | begin
refine le_hitting _ ω,
by_contra,
rw set.Icc_eq_empty_of_lt (not_le.mp h) at h_exists,
simpa using h_exists,
end | lemma | measure_theory.le_hitting_of_exists | probability.process | src/probability/process/hitting_time.lean | [
"probability.process.stopping"
] | [
"by_contra",
"set.Icc",
"set.Icc_eq_empty_of_lt"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hitting_mem_Icc {m : ι} (hnm : n ≤ m) (ω : Ω) : hitting u s n m ω ∈ set.Icc n m | ⟨le_hitting hnm ω, hitting_le ω⟩ | lemma | measure_theory.hitting_mem_Icc | probability.process | src/probability/process/hitting_time.lean | [
"probability.process.stopping"
] | [
"set.Icc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hitting_mem_set [is_well_order ι (<)] {m : ι} (h_exists : ∃ j ∈ set.Icc n m, u j ω ∈ s) :
u (hitting u s n m ω) ω ∈ s | begin
simp_rw [hitting, if_pos h_exists],
have h_nonempty : (set.Icc n m ∩ {i : ι | u i ω ∈ s}).nonempty,
{ obtain ⟨k, hk₁, hk₂⟩ := h_exists,
exact ⟨k, set.mem_inter hk₁ hk₂⟩, },
have h_mem := Inf_mem h_nonempty,
rw [set.mem_inter_iff] at h_mem,
exact h_mem.2,
end | lemma | measure_theory.hitting_mem_set | probability.process | src/probability/process/hitting_time.lean | [
"probability.process.stopping"
] | [
"Inf_mem",
"is_well_order",
"set.Icc",
"set.mem_inter",
"set.mem_inter_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hitting_mem_set_of_hitting_lt [is_well_order ι (<)] {m : ι}
(hl : hitting u s n m ω < m) :
u (hitting u s n m ω) ω ∈ s | begin
by_cases h : ∃ j ∈ set.Icc n m, u j ω ∈ s,
{ exact hitting_mem_set h },
{ simp_rw [hitting, if_neg h] at hl,
exact false.elim (hl.ne rfl) }
end | lemma | measure_theory.hitting_mem_set_of_hitting_lt | probability.process | src/probability/process/hitting_time.lean | [
"probability.process.stopping"
] | [
"is_well_order",
"set.Icc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hitting_le_of_mem {m : ι} (hin : n ≤ i) (him : i ≤ m) (his : u i ω ∈ s) :
hitting u s n m ω ≤ i | begin
have h_exists : ∃ k ∈ set.Icc n m, u k ω ∈ s := ⟨i, ⟨hin, him⟩, his⟩,
simp_rw [hitting, if_pos h_exists],
exact cInf_le (bdd_below.inter_of_left bdd_below_Icc) (set.mem_inter ⟨hin, him⟩ his),
end | lemma | measure_theory.hitting_le_of_mem | probability.process | src/probability/process/hitting_time.lean | [
"probability.process.stopping"
] | [
"bdd_below.inter_of_left",
"bdd_below_Icc",
"cInf_le",
"set.Icc",
"set.mem_inter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hitting_le_iff_of_exists [is_well_order ι (<)] {m : ι}
(h_exists : ∃ j ∈ set.Icc n m, u j ω ∈ s) :
hitting u s n m ω ≤ i ↔ ∃ j ∈ set.Icc n i, u j ω ∈ s | begin
split; intro h',
{ exact ⟨hitting u s n m ω, ⟨le_hitting_of_exists h_exists, h'⟩, hitting_mem_set h_exists⟩, },
{ have h'' : ∃ k ∈ set.Icc n (min m i), u k ω ∈ s,
{ obtain ⟨k₁, hk₁_mem, hk₁_s⟩ := h_exists,
obtain ⟨k₂, hk₂_mem, hk₂_s⟩ := h',
refine ⟨min k₁ k₂, ⟨le_min hk₁_mem.1 hk₂_mem.1, min... | lemma | measure_theory.hitting_le_iff_of_exists | probability.process | src/probability/process/hitting_time.lean | [
"probability.process.stopping"
] | [
"is_well_order",
"min_le_min",
"min_rec'",
"set.Icc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hitting_le_iff_of_lt [is_well_order ι (<)] {m : ι} (i : ι) (hi : i < m) :
hitting u s n m ω ≤ i ↔ ∃ j ∈ set.Icc n i, u j ω ∈ s | begin
by_cases h_exists : ∃ j ∈ set.Icc n m, u j ω ∈ s,
{ rw hitting_le_iff_of_exists h_exists, },
{ simp_rw [hitting, if_neg h_exists],
push_neg at h_exists,
simp only [not_le.mpr hi, set.mem_Icc, false_iff, not_exists, and_imp],
exact λ k hkn hki, h_exists k ⟨hkn, hki.trans hi.le⟩, },
end | lemma | measure_theory.hitting_le_iff_of_lt | probability.process | src/probability/process/hitting_time.lean | [
"probability.process.stopping"
] | [
"and_imp",
"is_well_order",
"not_exists",
"set.Icc",
"set.mem_Icc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hitting_lt_iff [is_well_order ι (<)] {m : ι} (i : ι) (hi : i ≤ m) :
hitting u s n m ω < i ↔ ∃ j ∈ set.Ico n i, u j ω ∈ s | begin
split; intro h',
{ have h : ∃ j ∈ set.Icc n m, u j ω ∈ s,
{ by_contra,
simp_rw [hitting, if_neg h, ← not_le] at h',
exact h' hi, },
exact ⟨hitting u s n m ω, ⟨le_hitting_of_exists h, h'⟩, hitting_mem_set h⟩, },
{ obtain ⟨k, hk₁, hk₂⟩ := h',
refine lt_of_le_of_lt _ hk₁.2,
exact hi... | lemma | measure_theory.hitting_lt_iff | probability.process | src/probability/process/hitting_time.lean | [
"probability.process.stopping"
] | [
"by_contra",
"is_well_order",
"set.Icc",
"set.Ico"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hitting_eq_hitting_of_exists
{m₁ m₂ : ι} (h : m₁ ≤ m₂) (h' : ∃ j ∈ set.Icc n m₁, u j ω ∈ s) :
hitting u s n m₁ ω = hitting u s n m₂ ω | begin
simp only [hitting, if_pos h'],
obtain ⟨j, hj₁, hj₂⟩ := h',
rw if_pos,
{ refine le_antisymm _ (cInf_le_cInf bdd_below_Icc.inter_of_left ⟨j, hj₁, hj₂⟩
(set.inter_subset_inter_left _ (set.Icc_subset_Icc_right h))),
refine le_cInf ⟨j, set.Icc_subset_Icc_right h hj₁, hj₂⟩ (λ i hi, _),
by_cases h... | lemma | measure_theory.hitting_eq_hitting_of_exists | probability.process | src/probability/process/hitting_time.lean | [
"probability.process.stopping"
] | [
"cInf_le",
"cInf_le_cInf",
"le_cInf",
"le_rfl",
"set.Icc",
"set.Icc_subset_Icc_right",
"set.inter_subset_inter_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hitting_mono {m₁ m₂ : ι} (hm : m₁ ≤ m₂) :
hitting u s n m₁ ω ≤ hitting u s n m₂ ω | begin
by_cases h : ∃ j ∈ set.Icc n m₁, u j ω ∈ s,
{ exact (hitting_eq_hitting_of_exists hm h).le },
{ simp_rw [hitting, if_neg h],
split_ifs with h',
{ obtain ⟨j, hj₁, hj₂⟩ := h',
refine le_cInf ⟨j, hj₁, hj₂⟩ _,
by_contra hneg, push_neg at hneg,
obtain ⟨i, hi₁, hi₂⟩ := hneg,
exact ... | lemma | measure_theory.hitting_mono | probability.process | src/probability/process/hitting_time.lean | [
"probability.process.stopping"
] | [
"by_contra",
"le_cInf",
"set.Icc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hitting_is_stopping_time
[conditionally_complete_linear_order ι] [is_well_order ι (<)] [countable ι]
[topological_space β] [pseudo_metrizable_space β] [measurable_space β] [borel_space β]
{f : filtration ι m} {u : ι → Ω → β} {s : set β} {n n' : ι}
(hu : adapted f u) (hs : measurable_set s) :
is_stopping_time ... | begin
intro i,
cases le_or_lt n' i with hi hi,
{ have h_le : ∀ ω, hitting u s n n' ω ≤ i := λ x, (hitting_le x).trans hi,
simp [h_le], },
{ have h_set_eq_Union : {ω | hitting u s n n' ω ≤ i} = ⋃ j ∈ set.Icc n i, u j ⁻¹' s,
{ ext x,
rw [set.mem_set_of_eq, hitting_le_iff_of_lt _ hi],
simp only... | lemma | measure_theory.hitting_is_stopping_time | probability.process | src/probability/process/hitting_time.lean | [
"probability.process.stopping"
] | [
"borel_space",
"conditionally_complete_linear_order",
"countable",
"exists_prop",
"is_well_order",
"measurable",
"measurable_set",
"measurable_set.Union",
"measurable_space",
"set.Icc",
"set.mem_Icc",
"set.mem_Union",
"set.mem_preimage",
"topological_space"
] | A discrete hitting time is a stopping time. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
stopped_value_hitting_mem [conditionally_complete_linear_order ι] [is_well_order ι (<)]
{u : ι → Ω → β} {s : set β} {n m : ι} {ω : Ω} (h : ∃ j ∈ set.Icc n m, u j ω ∈ s) :
stopped_value u (hitting u s n m) ω ∈ s | begin
simp only [stopped_value, hitting, if_pos h],
obtain ⟨j, hj₁, hj₂⟩ := h,
have : Inf (set.Icc n m ∩ {i | u i ω ∈ s}) ∈ set.Icc n m ∩ {i | u i ω ∈ s} :=
Inf_mem (set.nonempty_of_mem ⟨hj₁, hj₂⟩),
exact this.2,
end | lemma | measure_theory.stopped_value_hitting_mem | probability.process | src/probability/process/hitting_time.lean | [
"probability.process.stopping"
] | [
"Inf_mem",
"conditionally_complete_linear_order",
"is_well_order",
"set.Icc",
"set.nonempty_of_mem"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_stopping_time_hitting_is_stopping_time
[conditionally_complete_linear_order ι] [is_well_order ι (<)] [countable ι]
[topological_space ι] [order_topology ι] [first_countable_topology ι]
[topological_space β] [pseudo_metrizable_space β] [measurable_space β] [borel_space β]
{f : filtration ι m} {u : ι → Ω → β} ... | begin
intro n,
have h₁ : {x | hitting u s (τ x) N x ≤ n} =
(⋃ i ≤ n, {x | τ x = i} ∩ {x | hitting u s i N x ≤ n}) ∪
(⋃ i > n, {x | τ x = i} ∩ {x | hitting u s i N x ≤ n}),
{ ext x,
simp [← exists_or_distrib, ← or_and_distrib_right, le_or_lt] },
have h₂ : (⋃ i > n, {x | τ x = i} ∩ {x | hitting u s i ... | lemma | measure_theory.is_stopping_time_hitting_is_stopping_time | probability.process | src/probability/process/hitting_time.lean | [
"probability.process.stopping"
] | [
"borel_space",
"conditionally_complete_linear_order",
"countable",
"exists_or_distrib",
"exists_prop",
"gt_iff_lt",
"is_well_order",
"measurable_set",
"measurable_set.Union",
"measurable_space",
"not_and",
"not_exists",
"or_and_distrib_right",
"order_topology",
"set.mem_Union",
"set.me... | The hitting time of a discrete process with the starting time indexed by a stopping time
is a stopping time. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
hitting_eq_Inf (ω : Ω) : hitting u s ⊥ ⊤ ω = Inf {i : ι | u i ω ∈ s} | begin
simp only [hitting, set.mem_Icc, bot_le, le_top, and_self, exists_true_left, set.Icc_bot,
set.Iic_top, set.univ_inter, ite_eq_left_iff, not_exists],
intro h_nmem_s,
symmetry,
rw Inf_eq_top,
exact λ i hi_mem_s, absurd hi_mem_s (h_nmem_s i),
end | lemma | measure_theory.hitting_eq_Inf | probability.process | src/probability/process/hitting_time.lean | [
"probability.process.stopping"
] | [
"Inf_eq_top",
"bot_le",
"exists_true_left",
"ite_eq_left_iff",
"le_top",
"not_exists",
"set.Icc_bot",
"set.Iic_top",
"set.mem_Icc",
"set.univ_inter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hitting_bot_le_iff {i n : ι} {ω : Ω} (hx : ∃ j, j ≤ n ∧ u j ω ∈ s) :
hitting u s ⊥ n ω ≤ i ↔ ∃ j ≤ i, u j ω ∈ s | begin
cases lt_or_le i n with hi hi,
{ rw hitting_le_iff_of_lt _ hi,
simp, },
{ simp only [(hitting_le ω).trans hi, true_iff],
obtain ⟨j, hj₁, hj₂⟩ := hx,
exact ⟨j, hj₁.trans hi, hj₂⟩, },
end | lemma | measure_theory.hitting_bot_le_iff | probability.process | src/probability/process/hitting_time.lean | [
"probability.process.stopping"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_stopping_time [preorder ι] (f : filtration ι m) (τ : Ω → ι) | ∀ i : ι, measurable_set[f i] $ {ω | τ ω ≤ i} | def | measure_theory.is_stopping_time | probability.process | src/probability/process/stopping.lean | [
"probability.process.adapted"
] | [
"measurable_set"
] | A stopping time with respect to some filtration `f` is a function
`τ` such that for all `i`, the preimage of `{j | j ≤ i}` along `τ` is measurable
with respect to `f i`.
Intuitively, the stopping time `τ` describes some stopping rule such that at time
`i`, we may determine it with the information we have at time `i`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_stopping_time_const [preorder ι] (f : filtration ι m) (i : ι) :
is_stopping_time f (λ ω, i) | λ j, by simp only [measurable_set.const] | lemma | measure_theory.is_stopping_time_const | probability.process | src/probability/process/stopping.lean | [
"probability.process.adapted"
] | [
"measurable_set.const"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_stopping_time.measurable_set_le (hτ : is_stopping_time f τ) (i : ι) :
measurable_set[f i] {ω | τ ω ≤ i} | hτ i | lemma | measure_theory.is_stopping_time.measurable_set_le | probability.process | src/probability/process/stopping.lean | [
"probability.process.adapted"
] | [
"measurable_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_stopping_time.measurable_set_lt_of_pred [pred_order ι]
(hτ : is_stopping_time f τ) (i : ι) :
measurable_set[f i] {ω | τ ω < i} | begin
by_cases hi_min : is_min i,
{ suffices : {ω : Ω | τ ω < i} = ∅, by { rw this, exact @measurable_set.empty _ (f i), },
ext1 ω,
simp only [set.mem_set_of_eq, set.mem_empty_iff_false, iff_false],
rw is_min_iff_forall_not_lt at hi_min,
exact hi_min (τ ω), },
have : {ω : Ω | τ ω < i} = τ ⁻¹' (set... | lemma | measure_theory.is_stopping_time.measurable_set_lt_of_pred | probability.process | src/probability/process/stopping.lean | [
"probability.process.adapted"
] | [
"is_min",
"is_min_iff_forall_not_lt",
"measurable_set",
"measurable_set.empty",
"pred_order",
"set.Iio",
"set.mem_empty_iff_false"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
measurable_set_eq_of_countable_range
(hτ : is_stopping_time f τ) (h_countable : (set.range τ).countable) (i : ι) :
measurable_set[f i] {ω | τ ω = i} | begin
have : {ω | τ ω = i} = {ω | τ ω ≤ i} \ (⋃ (j ∈ set.range τ) (hj : j < i), {ω | τ ω ≤ j}),
{ ext1 a,
simp only [set.mem_set_of_eq, set.mem_range, set.Union_exists, set.Union_Union_eq',
set.mem_diff, set.mem_Union, exists_prop, not_exists, not_and, not_le],
split; intro h,
{ simp only [h, lt_i... | lemma | measure_theory.is_stopping_time.measurable_set_eq_of_countable_range | probability.process | src/probability/process/stopping.lean | [
"probability.process.adapted"
] | [
"and_imp",
"countable",
"exists_prop",
"imp_self",
"measurable_set",
"measurable_set.bUnion",
"measurable_set.empty",
"not_and",
"not_exists",
"set.Union_Union_eq'",
"set.Union_exists",
"set.Union_false",
"set.Union_true",
"set.mem_Union",
"set.mem_diff",
"set.mem_range",
"set.range"... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
measurable_set_eq_of_countable [countable ι] (hτ : is_stopping_time f τ) (i : ι) :
measurable_set[f i] {ω | τ ω = i} | hτ.measurable_set_eq_of_countable_range (set.to_countable _) i | lemma | measure_theory.is_stopping_time.measurable_set_eq_of_countable | probability.process | src/probability/process/stopping.lean | [
"probability.process.adapted"
] | [
"countable",
"measurable_set",
"set.to_countable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
measurable_set_lt_of_countable_range
(hτ : is_stopping_time f τ) (h_countable : (set.range τ).countable) (i : ι) :
measurable_set[f i] {ω | τ ω < i} | begin
have : {ω | τ ω < i} = {ω | τ ω ≤ i} \ {ω | τ ω = i},
{ ext1 ω, simp [lt_iff_le_and_ne], },
rw this,
exact (hτ.measurable_set_le i).diff (hτ.measurable_set_eq_of_countable_range h_countable i),
end | lemma | measure_theory.is_stopping_time.measurable_set_lt_of_countable_range | probability.process | src/probability/process/stopping.lean | [
"probability.process.adapted"
] | [
"countable",
"lt_iff_le_and_ne",
"measurable_set",
"set.range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
measurable_set_lt_of_countable [countable ι] (hτ : is_stopping_time f τ) (i : ι) :
measurable_set[f i] {ω | τ ω < i} | hτ.measurable_set_lt_of_countable_range (set.to_countable _) i | lemma | measure_theory.is_stopping_time.measurable_set_lt_of_countable | probability.process | src/probability/process/stopping.lean | [
"probability.process.adapted"
] | [
"countable",
"measurable_set",
"set.to_countable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
measurable_set_ge_of_countable_range {ι} [linear_order ι] {τ : Ω → ι}
{f : filtration ι m}
(hτ : is_stopping_time f τ) (h_countable : (set.range τ).countable) (i : ι) :
measurable_set[f i] {ω | i ≤ τ ω} | begin
have : {ω | i ≤ τ ω} = {ω | τ ω < i}ᶜ,
{ ext1 ω, simp only [set.mem_set_of_eq, set.mem_compl_iff, not_lt], },
rw this,
exact (hτ.measurable_set_lt_of_countable_range h_countable i).compl,
end | lemma | measure_theory.is_stopping_time.measurable_set_ge_of_countable_range | probability.process | src/probability/process/stopping.lean | [
"probability.process.adapted"
] | [
"countable",
"measurable_set",
"set.mem_compl_iff",
"set.range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
measurable_set_ge_of_countable {ι} [linear_order ι] {τ : Ω → ι} {f : filtration ι m}
[countable ι] (hτ : is_stopping_time f τ) (i : ι) :
measurable_set[f i] {ω | i ≤ τ ω} | hτ.measurable_set_ge_of_countable_range (set.to_countable _) i | lemma | measure_theory.is_stopping_time.measurable_set_ge_of_countable | probability.process | src/probability/process/stopping.lean | [
"probability.process.adapted"
] | [
"countable",
"measurable_set",
"set.to_countable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_stopping_time.measurable_set_gt (hτ : is_stopping_time f τ) (i : ι) :
measurable_set[f i] {ω | i < τ ω} | begin
have : {ω | i < τ ω} = {ω | τ ω ≤ i}ᶜ,
{ ext1 ω, simp only [set.mem_set_of_eq, set.mem_compl_iff, not_le], },
rw this,
exact (hτ.measurable_set_le i).compl,
end | lemma | measure_theory.is_stopping_time.measurable_set_gt | probability.process | src/probability/process/stopping.lean | [
"probability.process.adapted"
] | [
"measurable_set",
"set.mem_compl_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_stopping_time.measurable_set_lt_of_is_lub
(hτ : is_stopping_time f τ) (i : ι) (h_lub : is_lub (set.Iio i) i) :
measurable_set[f i] {ω | τ ω < i} | begin
by_cases hi_min : is_min i,
{ suffices : {ω | τ ω < i} = ∅, by { rw this, exact @measurable_set.empty _ (f i), },
ext1 ω,
simp only [set.mem_set_of_eq, set.mem_empty_iff_false, iff_false],
exact is_min_iff_forall_not_lt.mp hi_min (τ ω), },
obtain ⟨seq, -, -, h_tendsto, h_bound⟩ : ∃ seq : ℕ → ι,
... | lemma | measure_theory.is_stopping_time.measurable_set_lt_of_is_lub | probability.process | src/probability/process/stopping.lean | [
"probability.process.adapted"
] | [
"is_lub",
"is_min",
"is_open_Ioi",
"le_rfl",
"measurable_set",
"measurable_set.Union",
"measurable_set.empty",
"monotone",
"set.Ici",
"set.Iio",
"set.Ioi_subset_Ici",
"set.mem_Iio",
"set.mem_Union",
"set.mem_empty_iff_false",
"set.mem_preimage",
"set.preimage_Union",
"set.preimage_se... | Auxiliary lemma for `is_stopping_time.measurable_set_lt`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_stopping_time.measurable_set_lt (hτ : is_stopping_time f τ) (i : ι) :
measurable_set[f i] {ω | τ ω < i} | begin
obtain ⟨i', hi'_lub⟩ : ∃ i', is_lub (set.Iio i) i', from exists_lub_Iio i,
cases lub_Iio_eq_self_or_Iio_eq_Iic i hi'_lub with hi'_eq_i h_Iio_eq_Iic,
{ rw ← hi'_eq_i at hi'_lub ⊢,
exact hτ.measurable_set_lt_of_is_lub i' hi'_lub, },
{ have h_lt_eq_preimage : {ω : Ω | τ ω < i} = τ ⁻¹' (set.Iio i) := rfl,... | lemma | measure_theory.is_stopping_time.measurable_set_lt | probability.process | src/probability/process/stopping.lean | [
"probability.process.adapted"
] | [
"exists_lub_Iio",
"is_lub",
"lub_Iio_eq_self_or_Iio_eq_Iic",
"lub_Iio_le",
"measurable_set",
"set.Iio"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_stopping_time.measurable_set_ge (hτ : is_stopping_time f τ) (i : ι) :
measurable_set[f i] {ω | i ≤ τ ω} | begin
have : {ω | i ≤ τ ω} = {ω | τ ω < i}ᶜ,
{ ext1 ω, simp only [set.mem_set_of_eq, set.mem_compl_iff, not_lt], },
rw this,
exact (hτ.measurable_set_lt i).compl,
end | lemma | measure_theory.is_stopping_time.measurable_set_ge | probability.process | src/probability/process/stopping.lean | [
"probability.process.adapted"
] | [
"measurable_set",
"set.mem_compl_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_stopping_time.measurable_set_eq (hτ : is_stopping_time f τ) (i : ι) :
measurable_set[f i] {ω | τ ω = i} | begin
have : {ω | τ ω = i} = {ω | τ ω ≤ i} ∩ {ω | τ ω ≥ i},
{ ext1 ω, simp only [set.mem_set_of_eq, ge_iff_le, set.mem_inter_iff, le_antisymm_iff], },
rw this,
exact (hτ.measurable_set_le i).inter (hτ.measurable_set_ge i),
end | lemma | measure_theory.is_stopping_time.measurable_set_eq | probability.process | src/probability/process/stopping.lean | [
"probability.process.adapted"
] | [
"ge_iff_le",
"measurable_set",
"set.mem_inter_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_stopping_time.measurable_set_eq_le (hτ : is_stopping_time f τ) {i j : ι} (hle : i ≤ j) :
measurable_set[f j] {ω | τ ω = i} | f.mono hle _ $ hτ.measurable_set_eq i | lemma | measure_theory.is_stopping_time.measurable_set_eq_le | probability.process | src/probability/process/stopping.lean | [
"probability.process.adapted"
] | [
"measurable_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_stopping_time.measurable_set_lt_le (hτ : is_stopping_time f τ) {i j : ι} (hle : i ≤ j) :
measurable_set[f j] {ω | τ ω < i} | f.mono hle _ $ hτ.measurable_set_lt i | lemma | measure_theory.is_stopping_time.measurable_set_lt_le | probability.process | src/probability/process/stopping.lean | [
"probability.process.adapted"
] | [
"measurable_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_stopping_time_of_measurable_set_eq [preorder ι] [countable ι]
{f : filtration ι m} {τ : Ω → ι} (hτ : ∀ i, measurable_set[f i] {ω | τ ω = i}) :
is_stopping_time f τ | begin
intro i,
rw show {ω | τ ω ≤ i} = ⋃ k ≤ i, {ω | τ ω = k}, by { ext, simp },
refine measurable_set.bUnion (set.to_countable _) (λ k hk, _),
exact f.mono hk _ (hτ k),
end | lemma | measure_theory.is_stopping_time_of_measurable_set_eq | probability.process | src/probability/process/stopping.lean | [
"probability.process.adapted"
] | [
"countable",
"measurable_set",
"measurable_set.bUnion",
"set.to_countable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
max [linear_order ι] {f : filtration ι m} {τ π : Ω → ι}
(hτ : is_stopping_time f τ) (hπ : is_stopping_time f π) :
is_stopping_time f (λ ω, max (τ ω) (π ω)) | begin
intro i,
simp_rw [max_le_iff, set.set_of_and],
exact (hτ i).inter (hπ i),
end | lemma | measure_theory.is_stopping_time.max | probability.process | src/probability/process/stopping.lean | [
"probability.process.adapted"
] | [
"max_le_iff",
"set.set_of_and"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
max_const [linear_order ι] {f : filtration ι m} {τ : Ω → ι}
(hτ : is_stopping_time f τ) (i : ι) :
is_stopping_time f (λ ω, max (τ ω) i) | hτ.max (is_stopping_time_const f i) | lemma | measure_theory.is_stopping_time.max_const | probability.process | src/probability/process/stopping.lean | [
"probability.process.adapted"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
min [linear_order ι] {f : filtration ι m} {τ π : Ω → ι}
(hτ : is_stopping_time f τ) (hπ : is_stopping_time f π) :
is_stopping_time f (λ ω, min (τ ω) (π ω)) | begin
intro i,
simp_rw [min_le_iff, set.set_of_or],
exact (hτ i).union (hπ i),
end | lemma | measure_theory.is_stopping_time.min | probability.process | src/probability/process/stopping.lean | [
"probability.process.adapted"
] | [
"min_le_iff",
"set.set_of_or"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
min_const [linear_order ι] {f : filtration ι m} {τ : Ω → ι}
(hτ : is_stopping_time f τ) (i : ι) :
is_stopping_time f (λ ω, min (τ ω) i) | hτ.min (is_stopping_time_const f i) | lemma | measure_theory.is_stopping_time.min_const | probability.process | src/probability/process/stopping.lean | [
"probability.process.adapted"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_const [add_group ι] [preorder ι] [covariant_class ι ι (function.swap (+)) (≤)]
[covariant_class ι ι (+) (≤)]
{f : filtration ι m} {τ : Ω → ι} (hτ : is_stopping_time f τ) {i : ι} (hi : 0 ≤ i) :
is_stopping_time f (λ ω, τ ω + i) | begin
intro j,
simp_rw [← le_sub_iff_add_le],
exact f.mono (sub_le_self j hi) _ (hτ (j - i)),
end | lemma | measure_theory.is_stopping_time.add_const | probability.process | src/probability/process/stopping.lean | [
"probability.process.adapted"
] | [
"add_group",
"covariant_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_const_nat
{f : filtration ℕ m} {τ : Ω → ℕ} (hτ : is_stopping_time f τ) {i : ℕ} :
is_stopping_time f (λ ω, τ ω + i) | begin
refine is_stopping_time_of_measurable_set_eq (λ j, _),
by_cases hij : i ≤ j,
{ simp_rw [eq_comm, ← nat.sub_eq_iff_eq_add hij, eq_comm],
exact f.mono (j.sub_le i) _ (hτ.measurable_set_eq (j - i)) },
{ rw not_le at hij,
convert measurable_set.empty,
ext ω,
simp only [set.mem_empty_iff_false,... | lemma | measure_theory.is_stopping_time.add_const_nat | probability.process | src/probability/process/stopping.lean | [
"probability.process.adapted"
] | [
"measurable_set.empty",
"set.mem_empty_iff_false"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add
{f : filtration ℕ m} {τ π : Ω → ℕ} (hτ : is_stopping_time f τ) (hπ : is_stopping_time f π) :
is_stopping_time f (τ + π) | begin
intro i,
rw (_ : {ω | (τ + π) ω ≤ i} = ⋃ k ≤ i, {ω | π ω = k} ∩ {ω | τ ω + k ≤ i}),
{ exact measurable_set.Union (λ k, measurable_set.Union
(λ hk, (hπ.measurable_set_eq_le hk).inter (hτ.add_const_nat i))) },
ext ω,
simp only [pi.add_apply, set.mem_set_of_eq, set.mem_Union, set.mem_inter_iff, exist... | lemma | measure_theory.is_stopping_time.add | probability.process | src/probability/process/stopping.lean | [
"probability.process.adapted"
] | [
"exists_prop",
"measurable_set.Union",
"set.mem_Union",
"set.mem_inter_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
measurable_space (hτ : is_stopping_time f τ) : measurable_space Ω | { measurable_set' := λ s, ∀ i : ι, measurable_set[f i] (s ∩ {ω | τ ω ≤ i}),
measurable_set_empty :=
λ i, (set.empty_inter {ω | τ ω ≤ i}).symm ▸ @measurable_set.empty _ (f i),
measurable_set_compl := λ s hs i,
begin
rw (_ : sᶜ ∩ {ω | τ ω ≤ i} = (sᶜ ∪ {ω | τ ω ≤ i}ᶜ) ∩ {ω | τ ω ≤ i}),
{ refine mea... | def | measure_theory.is_stopping_time.measurable_space | probability.process | src/probability/process/stopping.lean | [
"probability.process.adapted"
] | [
"forall_swap",
"measurable_set",
"measurable_set.Union",
"measurable_set.empty",
"measurable_set.inter",
"measurable_space",
"set.Union_inter",
"set.compl_inter",
"set.compl_inter_self",
"set.empty_inter",
"set.union_empty",
"set.union_inter_distrib_right"
] | The associated σ-algebra with a stopping time. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
measurable_set (hτ : is_stopping_time f τ) (s : set Ω) :
measurable_set[hτ.measurable_space] s ↔
∀ i : ι, measurable_set[f i] (s ∩ {ω | τ ω ≤ i}) | iff.rfl | lemma | measure_theory.is_stopping_time.measurable_set | probability.process | src/probability/process/stopping.lean | [
"probability.process.adapted"
] | [
"measurable_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
measurable_space_mono
(hτ : is_stopping_time f τ) (hπ : is_stopping_time f π) (hle : τ ≤ π) :
hτ.measurable_space ≤ hπ.measurable_space | begin
intros s hs i,
rw (_ : s ∩ {ω | π ω ≤ i} = s ∩ {ω | τ ω ≤ i} ∩ {ω | π ω ≤ i}),
{ exact (hs i).inter (hπ i) },
{ ext,
simp only [set.mem_inter_iff, iff_self_and, and.congr_left_iff, set.mem_set_of_eq],
intros hle' _,
exact le_trans (hle _) hle' },
end | lemma | measure_theory.is_stopping_time.measurable_space_mono | probability.process | src/probability/process/stopping.lean | [
"probability.process.adapted"
] | [
"and.congr_left_iff",
"iff_self_and",
"set.mem_inter_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
measurable_space_le_of_countable [countable ι] (hτ : is_stopping_time f τ) :
hτ.measurable_space ≤ m | begin
intros s hs,
change ∀ i, measurable_set[f i] (s ∩ {ω | τ ω ≤ i}) at hs,
rw (_ : s = ⋃ i, s ∩ {ω | τ ω ≤ i}),
{ exact measurable_set.Union (λ i, f.le i _ (hs i)) },
{ ext ω, split; rw set.mem_Union,
{ exact λ hx, ⟨τ ω, hx, le_rfl⟩ },
{ rintro ⟨_, hx, _⟩,
exact hx } }
end | lemma | measure_theory.is_stopping_time.measurable_space_le_of_countable | probability.process | src/probability/process/stopping.lean | [
"probability.process.adapted"
] | [
"countable",
"measurable_set",
"measurable_set.Union",
"set.mem_Union"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
measurable_space_le' [is_countably_generated (at_top : filter ι)] [(at_top : filter ι).ne_bot]
(hτ : is_stopping_time f τ) :
hτ.measurable_space ≤ m | begin
intros s hs,
change ∀ i, measurable_set[f i] (s ∩ {ω | τ ω ≤ i}) at hs,
obtain ⟨seq : ℕ → ι, h_seq_tendsto⟩ := at_top.exists_seq_tendsto,
rw (_ : s = ⋃ n, s ∩ {ω | τ ω ≤ seq n}),
{ exact measurable_set.Union (λ i, f.le (seq i) _ (hs (seq i))), },
{ ext ω, split; rw set.mem_Union,
{ intros hx,
... | lemma | measure_theory.is_stopping_time.measurable_space_le' | probability.process | src/probability/process/stopping.lean | [
"probability.process.adapted"
] | [
"filter",
"measurable_set",
"measurable_set.Union",
"set.mem_Union"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
measurable_space_le {ι} [semilattice_sup ι] {f : filtration ι m} {τ : Ω → ι}
[is_countably_generated (at_top : filter ι)] (hτ : is_stopping_time f τ) :
hτ.measurable_space ≤ m | begin
casesI is_empty_or_nonempty ι,
{ haveI : is_empty Ω := ⟨λ ω, is_empty.false (τ ω)⟩,
intros s hsτ,
suffices hs : s = ∅, by { rw hs, exact measurable_set.empty, },
haveI : unique (set Ω) := set.unique_empty,
rw [unique.eq_default s, unique.eq_default ∅], },
exact measurable_space_le' hτ,
end | lemma | measure_theory.is_stopping_time.measurable_space_le | probability.process | src/probability/process/stopping.lean | [
"probability.process.adapted"
] | [
"filter",
"is_empty",
"is_empty_or_nonempty",
"measurable_set.empty",
"semilattice_sup",
"set.unique_empty",
"unique",
"unique.eq_default"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
measurable_space_const (f : filtration ι m) (i : ι) :
(is_stopping_time_const f i).measurable_space = f i | begin
ext1 s,
change measurable_set[(is_stopping_time_const f i).measurable_space] s ↔ measurable_set[f i] s,
rw is_stopping_time.measurable_set,
split; intro h,
{ specialize h i,
simpa only [le_refl, set.set_of_true, set.inter_univ] using h, },
{ intro j,
by_cases hij : i ≤ j,
{ simp only [hij,... | lemma | measure_theory.is_stopping_time.measurable_space_const | probability.process | src/probability/process/stopping.lean | [
"probability.process.adapted"
] | [
"measurable_set",
"measurable_set.empty",
"measurable_space",
"set.inter_empty",
"set.inter_univ",
"set.set_of_false",
"set.set_of_true"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
measurable_set_inter_eq_iff (hτ : is_stopping_time f τ) (s : set Ω) (i : ι) :
measurable_set[hτ.measurable_space] (s ∩ {ω | τ ω = i})
↔ measurable_set[f i] (s ∩ {ω | τ ω = i}) | begin
have : ∀ j, ({ω : Ω | τ ω = i} ∩ {ω : Ω | τ ω ≤ j}) = {ω : Ω | τ ω = i} ∩ {ω | i ≤ j},
{ intro j,
ext1 ω,
simp only [set.mem_inter_iff, set.mem_set_of_eq, and.congr_right_iff],
intro hxi,
rw hxi, },
split; intro h,
{ specialize h i,
simpa only [set.inter_assoc, this, le_refl, set.set_o... | lemma | measure_theory.is_stopping_time.measurable_set_inter_eq_iff | probability.process | src/probability/process/stopping.lean | [
"probability.process.adapted"
] | [
"and.congr_right_iff",
"measurable_set",
"set.inter_assoc",
"set.inter_univ",
"set.mem_inter_iff",
"set.set_of_true"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
measurable_space_le_of_le_const (hτ : is_stopping_time f τ) {i : ι} (hτ_le : ∀ ω, τ ω ≤ i) :
hτ.measurable_space ≤ f i | (measurable_space_mono hτ _ hτ_le).trans (measurable_space_const _ _).le | lemma | measure_theory.is_stopping_time.measurable_space_le_of_le_const | probability.process | src/probability/process/stopping.lean | [
"probability.process.adapted"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
measurable_space_le_of_le (hτ : is_stopping_time f τ) {n : ι} (hτ_le : ∀ ω, τ ω ≤ n) :
hτ.measurable_space ≤ m | (hτ.measurable_space_le_of_le_const hτ_le).trans (f.le n) | lemma | measure_theory.is_stopping_time.measurable_space_le_of_le | probability.process | src/probability/process/stopping.lean | [
"probability.process.adapted"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_measurable_space_of_const_le (hτ : is_stopping_time f τ) {i : ι} (hτ_le : ∀ ω, i ≤ τ ω) :
f i ≤ hτ.measurable_space | (measurable_space_const _ _).symm.le.trans (measurable_space_mono _ hτ hτ_le) | lemma | measure_theory.is_stopping_time.le_measurable_space_of_const_le | probability.process | src/probability/process/stopping.lean | [
"probability.process.adapted"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sigma_finite_stopping_time {ι} [semilattice_sup ι] [order_bot ι]
[(filter.at_top : filter ι).is_countably_generated]
{μ : measure Ω} {f : filtration ι m} {τ : Ω → ι}
[sigma_finite_filtration μ f] (hτ : is_stopping_time f τ) :
sigma_finite (μ.trim hτ.measurable_space_le) | begin
refine sigma_finite_trim_mono hτ.measurable_space_le _,
{ exact f ⊥, },
{ exact hτ.le_measurable_space_of_const_le (λ _, bot_le), },
{ apply_instance, },
end | instance | measure_theory.is_stopping_time.sigma_finite_stopping_time | probability.process | src/probability/process/stopping.lean | [
"probability.process.adapted"
] | [
"bot_le",
"filter",
"filter.at_top",
"order_bot",
"semilattice_sup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sigma_finite_stopping_time_of_le {ι} [semilattice_sup ι] [order_bot ι]
{μ : measure Ω} {f : filtration ι m} {τ : Ω → ι}
[sigma_finite_filtration μ f] (hτ : is_stopping_time f τ) {n : ι} (hτ_le : ∀ ω, τ ω ≤ n) :
sigma_finite (μ.trim (hτ.measurable_space_le_of_le hτ_le)) | begin
refine sigma_finite_trim_mono (hτ.measurable_space_le_of_le hτ_le) _,
{ exact f ⊥, },
{ exact hτ.le_measurable_space_of_const_le (λ _, bot_le), },
{ apply_instance, },
end | instance | measure_theory.is_stopping_time.sigma_finite_stopping_time_of_le | probability.process | src/probability/process/stopping.lean | [
"probability.process.adapted"
] | [
"bot_le",
"order_bot",
"semilattice_sup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
measurable_set_le' (hτ : is_stopping_time f τ) (i : ι) :
measurable_set[hτ.measurable_space] {ω | τ ω ≤ i} | begin
intro j,
have : {ω : Ω | τ ω ≤ i} ∩ {ω : Ω | τ ω ≤ j} = {ω : Ω | τ ω ≤ min i j},
{ ext1 ω, simp only [set.mem_inter_iff, set.mem_set_of_eq, le_min_iff], },
rw this,
exact f.mono (min_le_right i j) _ (hτ _),
end | lemma | measure_theory.is_stopping_time.measurable_set_le' | probability.process | src/probability/process/stopping.lean | [
"probability.process.adapted"
] | [
"le_min_iff",
"measurable_set",
"measurable_set_le'",
"set.mem_inter_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
measurable_set_gt' (hτ : is_stopping_time f τ) (i : ι) :
measurable_set[hτ.measurable_space] {ω | i < τ ω} | begin
have : {ω : Ω | i < τ ω} = {ω : Ω | τ ω ≤ i}ᶜ, by { ext1 ω, simp, },
rw this,
exact (hτ.measurable_set_le' i).compl,
end | lemma | measure_theory.is_stopping_time.measurable_set_gt' | probability.process | src/probability/process/stopping.lean | [
"probability.process.adapted"
] | [
"measurable_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
measurable_set_eq' [topological_space ι] [order_topology ι]
[first_countable_topology ι]
(hτ : is_stopping_time f τ) (i : ι) :
measurable_set[hτ.measurable_space] {ω | τ ω = i} | begin
rw [← set.univ_inter {ω | τ ω = i}, measurable_set_inter_eq_iff, set.univ_inter],
exact hτ.measurable_set_eq i,
end | lemma | measure_theory.is_stopping_time.measurable_set_eq' | probability.process | src/probability/process/stopping.lean | [
"probability.process.adapted"
] | [
"measurable_set",
"order_topology",
"set.univ_inter",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
measurable_set_ge' [topological_space ι] [order_topology ι]
[first_countable_topology ι]
(hτ : is_stopping_time f τ) (i : ι) :
measurable_set[hτ.measurable_space] {ω | i ≤ τ ω} | begin
have : {ω | i ≤ τ ω} = {ω | τ ω = i} ∪ {ω | i < τ ω},
{ ext1 ω,
simp only [le_iff_lt_or_eq, set.mem_set_of_eq, set.mem_union],
rw [@eq_comm _ i, or_comm], },
rw this,
exact (hτ.measurable_set_eq' i).union (hτ.measurable_set_gt' i),
end | lemma | measure_theory.is_stopping_time.measurable_set_ge' | probability.process | src/probability/process/stopping.lean | [
"probability.process.adapted"
] | [
"measurable_set",
"order_topology",
"set.mem_union",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
measurable_set_lt' [topological_space ι] [order_topology ι]
[first_countable_topology ι]
(hτ : is_stopping_time f τ) (i : ι) :
measurable_set[hτ.measurable_space] {ω | τ ω < i} | begin
have : {ω | τ ω < i} = {ω | τ ω ≤ i} \ {ω | τ ω = i},
{ ext1 ω,
simp only [lt_iff_le_and_ne, set.mem_set_of_eq, set.mem_diff], },
rw this,
exact (hτ.measurable_set_le' i).diff (hτ.measurable_set_eq' i),
end | lemma | measure_theory.is_stopping_time.measurable_set_lt' | probability.process | src/probability/process/stopping.lean | [
"probability.process.adapted"
] | [
"lt_iff_le_and_ne",
"measurable_set",
"measurable_set_lt'",
"order_topology",
"set.mem_diff",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
measurable_set_eq_of_countable_range'
(hτ : is_stopping_time f τ) (h_countable : (set.range τ).countable) (i : ι) :
measurable_set[hτ.measurable_space] {ω | τ ω = i} | begin
rw [← set.univ_inter {ω | τ ω = i}, measurable_set_inter_eq_iff, set.univ_inter],
exact hτ.measurable_set_eq_of_countable_range h_countable i,
end | lemma | measure_theory.is_stopping_time.measurable_set_eq_of_countable_range' | probability.process | src/probability/process/stopping.lean | [
"probability.process.adapted"
] | [
"countable",
"measurable_set",
"set.range",
"set.univ_inter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
measurable_set_eq_of_countable' [countable ι] (hτ : is_stopping_time f τ) (i : ι) :
measurable_set[hτ.measurable_space] {ω | τ ω = i} | hτ.measurable_set_eq_of_countable_range' (set.to_countable _) i | lemma | measure_theory.is_stopping_time.measurable_set_eq_of_countable' | probability.process | src/probability/process/stopping.lean | [
"probability.process.adapted"
] | [
"countable",
"measurable_set",
"set.to_countable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
measurable_set_ge_of_countable_range'
(hτ : is_stopping_time f τ) (h_countable : (set.range τ).countable) (i : ι) :
measurable_set[hτ.measurable_space] {ω | i ≤ τ ω} | begin
have : {ω | i ≤ τ ω} = {ω | τ ω = i} ∪ {ω | i < τ ω},
{ ext1 ω,
simp only [le_iff_lt_or_eq, set.mem_set_of_eq, set.mem_union],
rw [@eq_comm _ i, or_comm], },
rw this,
exact (hτ.measurable_set_eq_of_countable_range' h_countable i).union (hτ.measurable_set_gt' i),
end | lemma | measure_theory.is_stopping_time.measurable_set_ge_of_countable_range' | probability.process | src/probability/process/stopping.lean | [
"probability.process.adapted"
] | [
"countable",
"measurable_set",
"set.mem_union",
"set.range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
measurable_set_ge_of_countable' [countable ι] (hτ : is_stopping_time f τ) (i : ι) :
measurable_set[hτ.measurable_space] {ω | i ≤ τ ω} | hτ.measurable_set_ge_of_countable_range' (set.to_countable _) i | lemma | measure_theory.is_stopping_time.measurable_set_ge_of_countable' | probability.process | src/probability/process/stopping.lean | [
"probability.process.adapted"
] | [
"countable",
"measurable_set",
"set.to_countable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
measurable_set_lt_of_countable_range'
(hτ : is_stopping_time f τ) (h_countable : (set.range τ).countable) (i : ι) :
measurable_set[hτ.measurable_space] {ω | τ ω < i} | begin
have : {ω | τ ω < i} = {ω | τ ω ≤ i} \ {ω | τ ω = i},
{ ext1 ω,
simp only [lt_iff_le_and_ne, set.mem_set_of_eq, set.mem_diff], },
rw this,
exact (hτ.measurable_set_le' i).diff (hτ.measurable_set_eq_of_countable_range' h_countable i),
end | lemma | measure_theory.is_stopping_time.measurable_set_lt_of_countable_range' | probability.process | src/probability/process/stopping.lean | [
"probability.process.adapted"
] | [
"countable",
"lt_iff_le_and_ne",
"measurable_set",
"set.mem_diff",
"set.range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
measurable_set_lt_of_countable' [countable ι] (hτ : is_stopping_time f τ) (i : ι) :
measurable_set[hτ.measurable_space] {ω | τ ω < i} | hτ.measurable_set_lt_of_countable_range' (set.to_countable _) i | lemma | measure_theory.is_stopping_time.measurable_set_lt_of_countable' | probability.process | src/probability/process/stopping.lean | [
"probability.process.adapted"
] | [
"countable",
"measurable_set",
"set.to_countable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
measurable_space_le_of_countable_range (hτ : is_stopping_time f τ)
(h_countable : (set.range τ).countable) :
hτ.measurable_space ≤ m | begin
intros s hs,
change ∀ i, measurable_set[f i] (s ∩ {ω | τ ω ≤ i}) at hs,
rw (_ : s = ⋃ (i ∈ set.range τ), s ∩ {ω | τ ω ≤ i}),
{ exact measurable_set.bUnion h_countable (λ i _, f.le i _ (hs i)), },
{ ext ω,
split; rw set.mem_Union,
{ exact λ hx, ⟨τ ω, by simpa using hx⟩,},
{ rintro ⟨i, hx⟩,
... | lemma | measure_theory.is_stopping_time.measurable_space_le_of_countable_range | probability.process | src/probability/process/stopping.lean | [
"probability.process.adapted"
] | [
"countable",
"exists_and_distrib_right",
"exists_prop",
"measurable_set",
"measurable_set.bUnion",
"set.Union_exists",
"set.mem_Union",
"set.mem_inter_iff",
"set.mem_range",
"set.range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
measurable [topological_space ι] [measurable_space ι]
[borel_space ι] [order_topology ι] [second_countable_topology ι]
(hτ : is_stopping_time f τ) :
measurable[hτ.measurable_space] τ | @measurable_of_Iic ι Ω _ _ _ hτ.measurable_space _ _ _ _ (λ i, hτ.measurable_set_le' i) | lemma | measure_theory.is_stopping_time.measurable | probability.process | src/probability/process/stopping.lean | [
"probability.process.adapted"
] | [
"borel_space",
"measurable",
"measurable_of_Iic",
"measurable_space",
"order_topology",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
measurable_of_le [topological_space ι] [measurable_space ι]
[borel_space ι] [order_topology ι] [second_countable_topology ι]
(hτ : is_stopping_time f τ) {i : ι} (hτ_le : ∀ ω, τ ω ≤ i) :
measurable[f i] τ | hτ.measurable.mono (measurable_space_le_of_le_const _ hτ_le) le_rfl | lemma | measure_theory.is_stopping_time.measurable_of_le | probability.process | src/probability/process/stopping.lean | [
"probability.process.adapted"
] | [
"borel_space",
"le_rfl",
"measurable",
"measurable_space",
"order_topology",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
measurable_space_min (hτ : is_stopping_time f τ) (hπ : is_stopping_time f π) :
(hτ.min hπ).measurable_space = hτ.measurable_space ⊓ hπ.measurable_space | begin
refine le_antisymm _ _,
{ exact le_inf (measurable_space_mono _ hτ (λ _, min_le_left _ _))
(measurable_space_mono _ hπ (λ _, min_le_right _ _)), },
{ intro s,
change measurable_set[hτ.measurable_space] s ∧ measurable_set[hπ.measurable_space] s
→ measurable_set[(hτ.min hπ).measurable_space] s... | lemma | measure_theory.is_stopping_time.measurable_space_min | probability.process | src/probability/process/stopping.lean | [
"probability.process.adapted"
] | [
"le_inf",
"measurable_set",
"measurable_space",
"set.inter_union_distrib_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
measurable_set_min_iff (hτ : is_stopping_time f τ) (hπ : is_stopping_time f π) (s : set Ω) :
measurable_set[(hτ.min hπ).measurable_space] s
↔ measurable_set[hτ.measurable_space] s ∧ measurable_set[hπ.measurable_space] s | by { rw measurable_space_min, refl, } | lemma | measure_theory.is_stopping_time.measurable_set_min_iff | probability.process | src/probability/process/stopping.lean | [
"probability.process.adapted"
] | [
"measurable_set",
"measurable_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
measurable_space_min_const (hτ : is_stopping_time f τ) {i : ι} :
(hτ.min_const i).measurable_space = hτ.measurable_space ⊓ f i | by rw [hτ.measurable_space_min (is_stopping_time_const _ i), measurable_space_const] | lemma | measure_theory.is_stopping_time.measurable_space_min_const | probability.process | src/probability/process/stopping.lean | [
"probability.process.adapted"
] | [
"measurable_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
measurable_set_min_const_iff (hτ : is_stopping_time f τ) (s : set Ω)
{i : ι} :
measurable_set[(hτ.min_const i).measurable_space] s
↔ measurable_set[hτ.measurable_space] s ∧ measurable_set[f i] s | by rw [measurable_space_min_const, measurable_space.measurable_set_inf] | lemma | measure_theory.is_stopping_time.measurable_set_min_const_iff | probability.process | src/probability/process/stopping.lean | [
"probability.process.adapted"
] | [
"measurable_set",
"measurable_space",
"measurable_space.measurable_set_inf"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
measurable_set_inter_le [topological_space ι] [second_countable_topology ι] [order_topology ι]
[measurable_space ι] [borel_space ι]
(hτ : is_stopping_time f τ) (hπ : is_stopping_time f π) (s : set Ω)
(hs : measurable_set[hτ.measurable_space] s) :
measurable_set[(hτ.min hπ).measurable_space] (s ∩ {ω | τ ω ≤ π ω}... | begin
simp_rw is_stopping_time.measurable_set at ⊢ hs,
intro i,
have : (s ∩ {ω | τ ω ≤ π ω} ∩ {ω | min (τ ω) (π ω) ≤ i})
= (s ∩ {ω | τ ω ≤ i}) ∩ {ω | min (τ ω) (π ω) ≤ i} ∩ {ω | min (τ ω) i ≤ min (min (τ ω) (π ω)) i},
{ ext1 ω,
simp only [min_le_iff, set.mem_inter_iff, set.mem_set_of_eq, le_min_iff, le_... | lemma | measure_theory.is_stopping_time.measurable_set_inter_le | probability.process | src/probability/process/stopping.lean | [
"probability.process.adapted"
] | [
"and.congr_right_iff",
"and_imp",
"borel_space",
"le_min_iff",
"measurable_set",
"measurable_set_le",
"measurable_space",
"min_le_iff",
"not_and",
"order_topology",
"set.mem_inter_iff",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
measurable_set_inter_le_iff [topological_space ι]
[second_countable_topology ι] [order_topology ι] [measurable_space ι] [borel_space ι]
(hτ : is_stopping_time f τ) (hπ : is_stopping_time f π)
(s : set Ω) :
measurable_set[hτ.measurable_space] (s ∩ {ω | τ ω ≤ π ω})
↔ measurable_set[(hτ.min hπ).measurable_spac... | begin
split; intro h,
{ have : s ∩ {ω | τ ω ≤ π ω} = s ∩ {ω | τ ω ≤ π ω} ∩ {ω | τ ω ≤ π ω},
by rw [set.inter_assoc, set.inter_self],
rw this,
exact measurable_set_inter_le _ _ _ h, },
{ rw measurable_set_min_iff at h,
exact h.1, },
end | lemma | measure_theory.is_stopping_time.measurable_set_inter_le_iff | probability.process | src/probability/process/stopping.lean | [
"probability.process.adapted"
] | [
"borel_space",
"measurable_set",
"measurable_space",
"order_topology",
"set.inter_assoc",
"set.inter_self",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
measurable_set_inter_le_const_iff (hτ : is_stopping_time f τ) (s : set Ω) (i : ι) :
measurable_set[hτ.measurable_space] (s ∩ {ω | τ ω ≤ i})
↔ measurable_set[(hτ.min_const i).measurable_space] (s ∩ {ω | τ ω ≤ i}) | begin
rw [is_stopping_time.measurable_set_min_iff hτ (is_stopping_time_const _ i),
is_stopping_time.measurable_space_const, is_stopping_time.measurable_set],
refine ⟨λ h, ⟨h, _⟩, λ h j, h.1 j⟩,
specialize h i,
rwa [set.inter_assoc, set.inter_self] at h,
end | lemma | measure_theory.is_stopping_time.measurable_set_inter_le_const_iff | probability.process | src/probability/process/stopping.lean | [
"probability.process.adapted"
] | [
"measurable_set",
"measurable_space",
"set.inter_assoc",
"set.inter_self"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
measurable_set_le_stopping_time [topological_space ι]
[second_countable_topology ι] [order_topology ι] [measurable_space ι] [borel_space ι]
(hτ : is_stopping_time f τ) (hπ : is_stopping_time f π) :
measurable_set[hτ.measurable_space] {ω | τ ω ≤ π ω} | begin
rw hτ.measurable_set,
intro j,
have : {ω | τ ω ≤ π ω} ∩ {ω | τ ω ≤ j} = {ω | min (τ ω) j ≤ min (π ω) j} ∩ {ω | τ ω ≤ j},
{ ext1 ω,
simp only [set.mem_inter_iff, set.mem_set_of_eq, min_le_iff, le_min_iff, le_refl, and_true,
and.congr_left_iff],
intro h,
simp only [h, or_self, and_true],
... | lemma | measure_theory.is_stopping_time.measurable_set_le_stopping_time | probability.process | src/probability/process/stopping.lean | [
"probability.process.adapted"
] | [
"and.congr_left_iff",
"borel_space",
"le_min_iff",
"measurable_set",
"measurable_set.inter",
"measurable_set_le",
"measurable_space",
"min_le_iff",
"order_topology",
"set.mem_inter_iff",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.