Theorem: measurable_cSup
File: Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean

tactic: cases' eq_empty_or_nonempty s with h2s h2s
state_before: α : Type u_3
β : Type ?u.1642321
γ : Type ?u.1642324
γ₂ : Type ?u.1642327
δ : Type u_2
ι✝ : Sort y
s✝ t u : Set α
inst✝¹² : TopologicalSpace α
inst✝¹¹ : MeasurableSpace α
inst✝¹⁰ : BorelSpace α
inst✝⁹ : TopologicalSpace β
inst✝⁸ : MeasurableSpace β
inst✝⁷ : BorelSpace β
inst✝⁶ : TopologicalSpace γ
inst✝⁵ : MeasurableSpace γ
inst✝⁴ : BorelSpace γ
inst✝³ : MeasurableSpace δ
inst✝² : ConditionallyCompleteLinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
ι : Type u_1
f : ι → δ → α
s : Set ι
hs : Set.Countable s
hf : ∀ (i : ι), Measurable (f i)
bdd : ∀ (x : δ), BddAbove ((fun i => f i x) '' s)
⊢ Measurable fun x => sSup ((fun i => f i x) '' s)

tactic: simp [h2s, measurable_const]
state_before: case inl
α : Type u_3
β : Type ?u.1642321
γ : Type ?u.1642324
γ₂ : Type ?u.1642327
δ : Type u_2
ι✝ : Sort y
s✝ t u : Set α
inst✝¹² : TopologicalSpace α
inst✝¹¹ : MeasurableSpace α
inst✝¹⁰ : BorelSpace α
inst✝⁹ : TopologicalSpace β
inst✝⁸ : MeasurableSpace β
inst✝⁷ : BorelSpace β
inst✝⁶ : TopologicalSpace γ
inst✝⁵ : MeasurableSpace γ
inst✝⁴ : BorelSpace γ
inst✝³ : MeasurableSpace δ
inst✝² : ConditionallyCompleteLinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
ι : Type u_1
f : ι → δ → α
s : Set ι
hs : Set.Countable s
hf : ∀ (i : ι), Measurable (f i)
bdd : ∀ (x : δ), BddAbove ((fun i => f i x) '' s)
h2s : s = ∅
⊢ Measurable fun x => sSup ((fun i => f i x) '' s)

tactic: apply measurable_of_Iic
state_before: case inr
α : Type u_3
β : Type ?u.1642321
γ : Type ?u.1642324
γ₂ : Type ?u.1642327
δ : Type u_2
ι✝ : Sort y
s✝ t u : Set α
inst✝¹² : TopologicalSpace α
inst✝¹¹ : MeasurableSpace α
inst✝¹⁰ : BorelSpace α
inst✝⁹ : TopologicalSpace β
inst✝⁸ : MeasurableSpace β
inst✝⁷ : BorelSpace β
inst✝⁶ : TopologicalSpace γ
inst✝⁵ : MeasurableSpace γ
inst✝⁴ : BorelSpace γ
inst✝³ : MeasurableSpace δ
inst✝² : ConditionallyCompleteLinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
ι : Type u_1
f : ι → δ → α
s : Set ι
hs : Set.Countable s
hf : ∀ (i : ι), Measurable (f i)
bdd : ∀ (x : δ), BddAbove ((fun i => f i x) '' s)
h2s : Set.Nonempty s
⊢ Measurable fun x => sSup ((fun i => f i x) '' s)

tactic: intro y
state_before: case inr.hf
α : Type u_3
β : Type ?u.1642321
γ : Type ?u.1642324
γ₂ : Type ?u.1642327
δ : Type u_2
ι✝ : Sort y
s✝ t u : Set α
inst✝¹² : TopologicalSpace α
inst✝¹¹ : MeasurableSpace α
inst✝¹⁰ : BorelSpace α
inst✝⁹ : TopologicalSpace β
inst✝⁸ : MeasurableSpace β
inst✝⁷ : BorelSpace β
inst✝⁶ : TopologicalSpace γ
inst✝⁵ : MeasurableSpace γ
inst✝⁴ : BorelSpace γ
inst✝³ : MeasurableSpace δ
inst✝² : ConditionallyCompleteLinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
ι : Type u_1
f : ι → δ → α
s : Set ι
hs : Set.Countable s
hf : ∀ (i : ι), Measurable (f i)
bdd : ∀ (x : δ), BddAbove ((fun i => f i x) '' s)
h2s : Set.Nonempty s
⊢ ∀ (x : α), MeasurableSet ((fun x => sSup ((fun i => f i x) '' s)) ⁻¹' Iic x)

tactic: simp_rw [preimage, mem_Iic, csSup_le_iff (bdd _) (h2s.image _), ball_image_iff, setOf_forall]
state_before: case inr.hf
α : Type u_3
β : Type ?u.1642321
γ : Type ?u.1642324
γ₂ : Type ?u.1642327
δ : Type u_2
ι✝ : Sort y
s✝ t u : Set α
inst✝¹² : TopologicalSpace α
inst✝¹¹ : MeasurableSpace α
inst✝¹⁰ : BorelSpace α
inst✝⁹ : TopologicalSpace β
inst✝⁸ : MeasurableSpace β
inst✝⁷ : BorelSpace β
inst✝⁶ : TopologicalSpace γ
inst✝⁵ : MeasurableSpace γ
inst✝⁴ : BorelSpace γ
inst✝³ : MeasurableSpace δ
inst✝² : ConditionallyCompleteLinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
ι : Type u_1
f : ι → δ → α
s : Set ι
hs : Set.Countable s
hf : ∀ (i : ι), Measurable (f i)
bdd : ∀ (x : δ), BddAbove ((fun i => f i x) '' s)
h2s : Set.Nonempty s
y : α
⊢ MeasurableSet ((fun x => sSup ((fun i => f i x) '' s)) ⁻¹' Iic y)

tactic: exact MeasurableSet.biInter hs fun i _ => measurableSet_le (hf i) measurable_const
state_before: case inr.hf
α : Type u_3
β : Type ?u.1642321
γ : Type ?u.1642324
γ₂ : Type ?u.1642327
δ : Type u_2
ι✝ : Sort y
s✝ t u : Set α
inst✝¹² : TopologicalSpace α
inst✝¹¹ : MeasurableSpace α
inst✝¹⁰ : BorelSpace α
inst✝⁹ : TopologicalSpace β
inst✝⁸ : MeasurableSpace β
inst✝⁷ : BorelSpace β
inst✝⁶ : TopologicalSpace γ
inst✝⁵ : MeasurableSpace γ
inst✝⁴ : BorelSpace γ
inst✝³ : MeasurableSpace δ
inst✝² : ConditionallyCompleteLinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
ι : Type u_1
f : ι → δ → α
s : Set ι
hs : Set.Countable s
hf : ∀ (i : ι), Measurable (f i)
bdd : ∀ (x : δ), BddAbove ((fun i => f i x) '' s)
h2s : Set.Nonempty s
y : α
⊢ MeasurableSet (⋂ (i : ι) (_ : i ∈ s), {x | f i x ≤ y})