Module: Mathlib.Probability.ProductMeasure

Goal state before:
pp: ι : Type u_1
X : ι → Type u_2
mX : (i : ι) → MeasurableSpace (X i)
μ : (i : ι) → Measure (X i)
hμ : ∀ (i : ι), IsProbabilityMeasure (μ i)
α : Type u_3
e : α ≃ ι
s : Finset ι
t : (i : ι) → Set (X i)
ht : ∀ (i : ι), MeasurableSet (t i)
⊢ ∏ i ∈ s.preimage ⇑e ⋯, (μ (e i)) (t (e i)) = ∏ i ∈ s, (μ i) (t i)

pp: case mt
ι : Type u_1
X : ι → Type u_2
mX : (i : ι) → MeasurableSpace (X i)
μ : (i : ι) → Measure (X i)
hμ : ∀ (i : ι), IsProbabilityMeasure (μ i)
α : Type u_3
e : α ≃ ι
s : Finset ι
t : (i : ι) → Set (X i)
ht : ∀ (i : ι), MeasurableSet (t i)
⊢ ∀ i ∈ s.preimage ⇑e ⋯, MeasurableSet (t (e i))

pp: case hf
ι : Type u_1
X : ι → Type u_2
mX : (i : ι) → MeasurableSpace (X i)
μ : (i : ι) → Measure (X i)
hμ : ∀ (i : ι), IsProbabilityMeasure (μ i)
α : Type u_3
e : α ≃ ι
s : Finset ι
t : (i : ι) → Set (X i)
ht : ∀ (i : ι), MeasurableSet (t i)
⊢ Measurable ⇑(piCongrLeft X e)

pp: case hs
ι : Type u_1
X : ι → Type u_2
mX : (i : ι) → MeasurableSpace (X i)
μ : (i : ι) → Measure (X i)
hμ : ∀ (i : ι), IsProbabilityMeasure (μ i)
α : Type u_3
e : α ≃ ι
s : Finset ι
t : (i : ι) → Set (X i)
ht : ∀ (i : ι), MeasurableSet (t i)
⊢ MeasurableSet ((⇑e '' ↑(s.preimage ⇑e ⋯)).pi t)

Tactic:
prod_equiv e

Goal state after:
ι : Type u_1
X : ι → Type u_2
mX : (i : ι) → MeasurableSpace (X i)
μ : (i : ι) → Measure (X i)
hμ : ∀ (i : ι), IsProbabilityMeasure (μ i)
α : Type u_3
e : α ≃ ι
s : Finset ι
t : (i : ι) → Set (X i)
ht : ∀ (i : ι), MeasurableSet (t i)
⊢ ∏ i ∈ ?m.94, ?m.96 i = ∏ i ∈ s, (μ i) (t i)

case hst
ι : Type u_1
X : ι → Type u_2
mX : (i : ι) → MeasurableSpace (X i)
μ : (i : ι) → Measure (X i)
hμ : ∀ (i : ι), IsProbabilityMeasure (μ i)
α : Type u_3
e : α ≃ ι
s : Finset ι
t : (i : ι) → Set (X i)
ht : ∀ (i : ι), MeasurableSet (t i)
⊢ ∀ (i : α), i ∈ s.preimage ⇑e ⋯ ↔ e i ∈ ?m.94

case hfg
ι : Type u_1
X : ι → Type u_2
mX : (i : ι) → MeasurableSpace (X i)
μ : (i : ι) → Measure (X i)
hμ : ∀ (i : ι), IsProbabilityMeasure (μ i)
α : Type u_3
e : α ≃ ι
s : Finset ι
t : (i : ι) → Set (X i)
ht : ∀ (i : ι), MeasurableSet (t i)
⊢ ∀ i ∈ s.preimage ⇑e ⋯, (μ (e i)) (t (e i)) = ?m.96 (e i)

ι : Type u_1
X : ι → Type u_2
mX : (i : ι) → MeasurableSpace (X i)
μ : (i : ι) → Measure (X i)
hμ : ∀ (i : ι), IsProbabilityMeasure (μ i)
α : Type u_3
e : α ≃ ι
s : Finset ι
t : (i : ι) → Set (X i)
ht : ∀ (i : ι), MeasurableSet (t i)
⊢ Finset ι

ι : Type u_1
X : ι → Type u_2
mX : (i : ι) → MeasurableSpace (X i)
μ : (i : ι) → Measure (X i)
hμ : ∀ (i : ι), IsProbabilityMeasure (μ i)
α : Type u_3
e : α ≃ ι
s : Finset ι
t : (i : ι) → Set (X i)
ht : ∀ (i : ι), MeasurableSet (t i)
⊢ ι → ℝ≥0∞

case mt
ι : Type u_1
X : ι → Type u_2
mX : (i : ι) → MeasurableSpace (X i)
μ : (i : ι) → Measure (X i)
hμ : ∀ (i : ι), IsProbabilityMeasure (μ i)
α : Type u_3
e : α ≃ ι
s : Finset ι
t : (i : ι) → Set (X i)
ht : ∀ (i : ι), MeasurableSet (t i)
⊢ ∀ i ∈ s.preimage ⇑e ⋯, MeasurableSet (t (e i))

case hf
ι : Type u_1
X : ι → Type u_2
mX : (i : ι) → MeasurableSpace (X i)
μ : (i : ι) → Measure (X i)
hμ : ∀ (i : ι), IsProbabilityMeasure (μ i)
α : Type u_3
e : α ≃ ι
s : Finset ι
t : (i : ι) → Set (X i)
ht : ∀ (i : ι), MeasurableSet (t i)
⊢ Measurable ⇑(piCongrLeft X e)

case hs
ι : Type u_1
X : ι → Type u_2
mX : (i : ι) → MeasurableSpace (X i)
μ : (i : ι) → Measure (X i)
hμ : ∀ (i : ι), IsProbabilityMeasure (μ i)
α : Type u_3
e : α ≃ ι
s : Finset ι
t : (i : ι) → Set (X i)
ht : ∀ (i : ι), MeasurableSet (t i)
⊢ MeasurableSet ((⇑e '' ↑(s.preimage ⇑e ⋯)).pi t)