case h
α : Type u_2
β : Type u_3
inst✝² : MeasurableSpace α
inst✝¹ : MeasurableSpace β
γ : Type u_1
inst✝ : MeasurableSpace γ
m : Measure α
f : α → Measure β
g : β → Measure γ
hf : Measurable f
hg : Measurable g
s : Set γ
hs : MeasurableSet s
⊢ ↑↑(bind (bind m f) g) s = ↑↑(bind m fun a => bind (f a) g) s
α : Type u_2
β : Type u_3
inst✝² : MeasurableSpace α
inst✝¹ : MeasurableSpace β
γ : Type u_1
inst✝ : MeasurableSpace γ
m : Measure α
f : α → Measure β
g : β → Measure γ
hf : Measurable f
hg : Measurable g
⊢ bind (bind m f) g = bind m fun a => bind (f a) g
ext1 s hs
case h
α : Type u_2
β : Type u_3
inst✝² : MeasurableSpace α
inst✝¹ : MeasurableSpace β
γ : Type u_1
inst✝ : MeasurableSpace γ
m : Measure α
f : α → Measure β
g : β → Measure γ
hf : Measurable f
hg : Measurable g
s : Set γ
hs : MeasurableSet s
⊢ (∫⁻ (a : α), ∫⁻ (x : β), ((fun μ => ↑↑μ s) ∘ g) x ∂f a ∂m) = ∫⁻ (a : α), ↑↑(((fun m => bind m g) ∘ f) a) s ∂m
case h
α : Type u_2
β : Type u_3
inst✝² : MeasurableSpace α
inst✝¹ : MeasurableSpace β
γ : Type u_1
inst✝ : MeasurableSpace γ
m : Measure α
f : α → Measure β
g : β → Measure γ
hf : Measurable f
hg : Measurable g
s : Set γ
hs : MeasurableSet s
⊢ ↑↑(bind (bind m f) g) s = ↑↑(bind m fun a => bind (f a) g) s
erw [bind_apply hs hg, bind_apply hs ((measurable_bind' hg).comp hf),
  lintegral_bind hf ((measurable_coe hs).comp hg)]
no goals
case h
α : Type u_2
β : Type u_3
inst✝² : MeasurableSpace α
inst✝¹ : MeasurableSpace β
γ : Type u_1
inst✝ : MeasurableSpace γ
m : Measure α
f : α → Measure β
g : β → Measure γ
hf : Measurable f
hg : Measurable g
s : Set γ
hs : MeasurableSet s
⊢ (∫⁻ (a : α), ∫⁻ (x : β), ((fun μ => ↑↑μ s) ∘ g) x ∂f a ∂m) = ∫⁻ (a : α), ↑↑(((fun m => bind m g) ∘ f) a) s ∂m
conv_rhs => enter [2, a]; erw [bind_apply hs hg]