Theorem: MeasureTheory.measurableSet_tendsto_fun
File: Mathlib/MeasureTheory/Constructions/Polish/Basic.lean

Proof steps:
state_before: α✝ : Type u_1
ι : Type u_2
γ : Type u_3
α : Type u_4
β : Type u_5
tβ : TopologicalSpace β
inst✝⁹ : T2Space β
inst✝⁸ : MeasurableSpace β
inst✝⁷ : MeasurableSpace α
s : Set γ
f✝ : γ → β
inst✝⁶ : MeasurableSpace γ
inst✝⁵ : Countable ι
l : Filter ι
inst✝⁴ : l.IsCountablyGenerated
inst✝³ : TopologicalSpace γ
inst✝² : SecondCountableTopology γ
inst✝¹ : PseudoMetrizableSpace γ
inst✝ : OpensMeasurableSpace γ
f : ι → β → γ
hf : ∀ (i : ι), Measurable (f i)
g : β → γ
hg : Measurable g
⊢ MeasurableSet {x | Tendsto (fun n => f n x) l (𝓝 (g x))}
state_after: α✝ : Type u_1
ι : Type u_2
γ : Type u_3
α : Type u_4
β : Type u_5
tβ : TopologicalSpace β
inst✝⁹ : T2Space β
inst✝⁸ : MeasurableSpace β
inst✝⁷ : MeasurableSpace α
s : Set γ
f✝ : γ → β
inst✝⁶ : MeasurableSpace γ
inst✝⁵ : Countable ι
l : Filter ι
inst✝⁴ : l.IsCountablyGenerated
inst✝³ : TopologicalSpace γ
inst✝² : SecondCountableTopology γ
inst✝¹ : PseudoMetrizableSpace γ
inst✝ : OpensMeasurableSpace γ
f : ι → β → γ
hf : ∀ (i : ι), Measurable (f i)
g : β → γ
hg : Measurable g
this : PseudoMetricSpace γ := pseudoMetrizableSpacePseudoMetric γ
⊢ MeasurableSet {x | Tendsto (fun n => f n x) l (𝓝 (g x))}
tactic: letI := TopologicalSpace.pseudoMetrizableSpacePseudoMetric γ
state_before: α✝ : Type u_1
ι : Type u_2
γ : Type u_3
α : Type u_4
β : Type u_5
tβ : TopologicalSpace β
inst✝⁹ : T2Space β
inst✝⁸ : MeasurableSpace β
inst✝⁷ : MeasurableSpace α
s : Set γ
f✝ : γ → β
inst✝⁶ : MeasurableSpace γ
inst✝⁵ : Countable ι
l : Filter ι
inst✝⁴ : l.IsCountablyGenerated
inst✝³ : TopologicalSpace γ
inst✝² : SecondCountableTopology γ
inst✝¹ : PseudoMetrizableSpace γ
inst✝ : OpensMeasurableSpace γ
f : ι → β → γ
hf : ∀ (i : ι), Measurable (f i)
g : β → γ
hg : Measurable g
this : PseudoMetricSpace γ := pseudoMetrizableSpacePseudoMetric γ
⊢ MeasurableSet {x | Tendsto (fun n => f n x) l (𝓝 (g x))}
state_after: α✝ : Type u_1
ι : Type u_2
γ : Type u_3
α : Type u_4
β : Type u_5
tβ : TopologicalSpace β
inst✝⁹ : T2Space β
inst✝⁸ : MeasurableSpace β
inst✝⁷ : MeasurableSpace α
s : Set γ
f✝ : γ → β
inst✝⁶ : MeasurableSpace γ
inst✝⁵ : Countable ι
l : Filter ι
inst✝⁴ : l.IsCountablyGenerated
inst✝³ : TopologicalSpace γ
inst✝² : SecondCountableTopology γ
inst✝¹ : PseudoMetrizableSpace γ
inst✝ : OpensMeasurableSpace γ
f : ι → β → γ
hf : ∀ (i : ι), Measurable (f i)
g : β → γ
hg : Measurable g
this : PseudoMetricSpace γ := pseudoMetrizableSpacePseudoMetric γ
⊢ MeasurableSet {x | Tendsto (fun b => dist (f b x) (g x)) l (𝓝 0)}
tactic: simp_rw [tendsto_iff_dist_tendsto_zero (f := fun n ↦ f n _)]
state_before: α✝ : Type u_1
ι : Type u_2
γ : Type u_3
α : Type u_4
β : Type u_5
tβ : TopologicalSpace β
inst✝⁹ : T2Space β
inst✝⁸ : MeasurableSpace β
inst✝⁷ : MeasurableSpace α
s : Set γ
f✝ : γ → β
inst✝⁶ : MeasurableSpace γ
inst✝⁵ : Countable ι
l : Filter ι
inst✝⁴ : l.IsCountablyGenerated
inst✝³ : TopologicalSpace γ
inst✝² : SecondCountableTopology γ
inst✝¹ : PseudoMetrizableSpace γ
inst✝ : OpensMeasurableSpace γ
f : ι → β → γ
hf : ∀ (i : ι), Measurable (f i)
g : β → γ
hg : Measurable g
this : PseudoMetricSpace γ := pseudoMetrizableSpacePseudoMetric γ
⊢ MeasurableSet {x | Tendsto (fun b => dist (f b x) (g x)) l (𝓝 0)}
state_after: no goals
tactic: exact measurableSet_tendsto (𝓝 0) (fun n ↦ (hf n).dist hg)