File: Mathlib/MeasureTheory/Function/AEMeasurableSequence.lean

full_name: aeSeqSet
code: /-- If we have the additional hypothesis `∀ᵐ x ∂μ, p x (fun n ↦ f n x)`, this is a measurable set
whose complement has measure 0 such that for all `x ∈ aeSeqSet`, `f i x` is equal to
`(hf i).mk (f i) x` for all `i` and we have the pointwise property `p x (fun n ↦ f n x)`. -/
def aeSeqSet (hf : ∀ i, AEMeasurable (f i) μ) (p : α → (ι → β) → Prop) : Set α 
goal: Set α

full_name: aeSeq
code: /-- A sequence of measurable functions that are equal to `f` and verify property `p` on the
measurable set `aeSeqSet hf p`. -/
noncomputable def aeSeq (hf : ∀ i, AEMeasurable (f i) μ) (p : α → (ι → β) → Prop) : ι → α → β 
goal: ι → α → β

full_name: aeSeq.mk_eq_fun_of_mem_aeSeqSet
code: theorem mk_eq_fun_of_mem_aeSeqSet (hf : ∀ i, AEMeasurable (f i) μ) {x : α} (hx : x ∈ aeSeqSet hf p)
    (i : ι) : (hf i).mk (f i) x = f i x 
goal: AEMeasurable.mk (f i) ⋯ x = f i x

full_name: aeSeq.aeSeq_eq_mk_of_mem_aeSeqSet
code: theorem aeSeq_eq_mk_of_mem_aeSeqSet (hf : ∀ i, AEMeasurable (f i) μ) {x : α}
    (hx : x ∈ aeSeqSet hf p) (i : ι) : aeSeq hf p i x = (hf i).mk (f i) x 
goal: aeSeq hf p i x = AEMeasurable.mk (f i) ⋯ x

full_name: aeSeq.aeSeq_eq_fun_of_mem_aeSeqSet
code: theorem aeSeq_eq_fun_of_mem_aeSeqSet (hf : ∀ i, AEMeasurable (f i) μ) {x : α}
    (hx : x ∈ aeSeqSet hf p) (i : ι) : aeSeq hf p i x = f i x 
goal: aeSeq hf p i x = f i x

full_name: aeSeq.prop_of_mem_aeSeqSet
code: theorem prop_of_mem_aeSeqSet (hf : ∀ i, AEMeasurable (f i) μ) {x : α} (hx : x ∈ aeSeqSet hf p) :
    p x fun n => aeSeq hf p n x 
goal: p x fun n => aeSeq hf p n x

full_name: aeSeq.fun_prop_of_mem_aeSeqSet
code: theorem fun_prop_of_mem_aeSeqSet (hf : ∀ i, AEMeasurable (f i) μ) {x : α} (hx : x ∈ aeSeqSet hf p) :
    p x fun n => f n x 
goal: p x fun n => f n x

full_name: aeSeq.aeSeqSet_measurableSet
code: theorem aeSeqSet_measurableSet {hf : ∀ i, AEMeasurable (f i) μ} : MeasurableSet (aeSeqSet hf p) 
goal: MeasurableSet (aeSeqSet hf p)

full_name: aeSeq.measurable
code: theorem measurable (hf : ∀ i, AEMeasurable (f i) μ) (p : α → (ι → β) → Prop) (i : ι) :
    Measurable (aeSeq hf p i) 
goal: Measurable (aeSeq hf p i)

full_name: aeSeq.measure_compl_aeSeqSet_eq_zero
code: theorem measure_compl_aeSeqSet_eq_zero [Countable ι] (hf : ∀ i, AEMeasurable (f i) μ)
    (hp : ∀ᵐ x ∂μ, p x fun n => f n x) : μ (aeSeqSet hf p)ᶜ = 0 
goal: μ (aeSeqSet hf p)ᶜ = 0

full_name: aeSeq.aeSeq_eq_mk_ae
code: theorem aeSeq_eq_mk_ae [Countable ι] (hf : ∀ i, AEMeasurable (f i) μ)
    (hp : ∀ᵐ x ∂μ, p x fun n => f n x) : ∀ᵐ a : α ∂μ, ∀ i : ι, aeSeq hf p i a = (hf i).mk (f i) a 
goal: ∀ᵐ (a : α) ∂μ, ∀ (i : ι), aeSeq hf p i a = AEMeasurable.mk (f i) ⋯ a

full_name: aeSeq.aeSeq_eq_fun_ae
code: theorem aeSeq_eq_fun_ae [Countable ι] (hf : ∀ i, AEMeasurable (f i) μ)
    (hp : ∀ᵐ x ∂μ, p x fun n => f n x) : ∀ᵐ a : α ∂μ, ∀ i : ι, aeSeq hf p i a = f i a 
goal: ∀ᵐ (a : α) ∂μ, ∀ (i : ι), aeSeq hf p i a = f i a

full_name: aeSeq.aeSeq_n_eq_fun_n_ae
code: theorem aeSeq_n_eq_fun_n_ae [Countable ι] (hf : ∀ i, AEMeasurable (f i) μ)
    (hp : ∀ᵐ x ∂μ, p x fun n => f n x) (n : ι) : aeSeq hf p n =ᵐ[μ] f n 
goal: aeSeq hf p n =ᵐ[μ] f n

full_name: aeSeq.iSup
code: theorem iSup [CompleteLattice β] [Countable ι] (hf : ∀ i, AEMeasurable (f i) μ)
    (hp : ∀ᵐ x ∂μ, p x fun n => f n x) : ⨆ n, aeSeq hf p n =ᵐ[μ] ⨆ n, f n 
goal: ⨆ n, aeSeq hf p n =ᵐ[μ] ⨆ n, f n

full_name: aeSeq.iInf
code: theorem iInf [CompleteLattice β] [Countable ι] (hf : ∀ i, AEMeasurable (f i) μ)
    (hp : ∀ᵐ x ∂μ, p x fun n ↦ f n x) : ⨅ n, aeSeq hf p n =ᵐ[μ] ⨅ n, f n 
goal: ⨅ n, aeSeq hf p n =ᵐ[μ] ⨅ n, f n