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refactor(MeasureTheory): golf Mathlib/MeasureTheory/Function/SimpleFunc #38492

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refactor(MeasureTheory): golf
yuanyi-350 Apr 25, 2026
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Update SimpleFunc.lean
yuanyi-350 Apr 25, 2026
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44 changes: 4 additions & 40 deletions Mathlib/MeasureTheory/Function/SimpleFunc.lean
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Original file line number Diff line number Diff line change
Expand Up @@ -1326,51 +1326,15 @@ protected theorem induction' {α γ} [MeasurableSpace α] [Nonempty γ] {P : Sim
measurable. -/
theorem _root_.Measurable.add_simpleFunc
{E : Type*} {_ : MeasurableSpace α} [MeasurableSpace E] [AddCancelMonoid E] [MeasurableAdd E]
{g : α → E} (hg : Measurable g) (f : SimpleFunc α E) :
Measurable (g + (f : α → E)) := by
classical
induction f using SimpleFunc.induction with
| @const c s hs =>
simp only [SimpleFunc.const_zero, SimpleFunc.coe_piecewise, SimpleFunc.coe_const,
SimpleFunc.coe_zero]
rw [← s.piecewise_same g, ← piecewise_add]
exact Measurable.piecewise hs (hg.add_const _) (hg.add_const _)
| @add f f' hff' hf hf' =>
have : (g + ↑(f + f')) = (Function.support f).piecewise (g + (f : α → E)) (g + f') := by
ext x
by_cases hx : x ∈ Function.support f
· simpa only [SimpleFunc.coe_add, Pi.add_apply, Function.mem_support, ne_eq, not_not,
Set.piecewise_eq_of_mem _ _ _ hx, _root_.add_right_inj, add_eq_left]
using Set.disjoint_left.1 hff' hx
· simpa only [SimpleFunc.coe_add, Pi.add_apply, Function.mem_support, ne_eq, not_not,
Set.piecewise_eq_of_notMem _ _ _ hx, _root_.add_right_inj, add_eq_right] using hx
rw [this]
exact Measurable.piecewise f.measurableSet_support hf hf'
{g : α → E} (hg : Measurable g) (f : SimpleFunc α E) : Measurable (g + (f : α → E)) :=
f.measurable_bind (fun b a ↦ g a + b) fun b ↦ hg.add_const b

/-- In a topological vector space, the addition of a simple function and a measurable function is
measurable. -/
theorem _root_.Measurable.simpleFunc_add
{E : Type*} {_ : MeasurableSpace α} [MeasurableSpace E] [AddCancelMonoid E] [MeasurableAdd E]
{g : α → E} (hg : Measurable g) (f : SimpleFunc α E) :
Measurable ((f : α → E) + g) := by
classical
induction f using SimpleFunc.induction with
| @const c s hs =>
simp only [SimpleFunc.const_zero, SimpleFunc.coe_piecewise, SimpleFunc.coe_const,
SimpleFunc.coe_zero]
rw [← s.piecewise_same g, ← piecewise_add]
exact Measurable.piecewise hs (hg.const_add _) (hg.const_add _)
| @add f f' hff' hf hf' =>
have : (↑(f + f') + g) = (Function.support f).piecewise ((f : α → E) + g) (f' + g) := by
ext x
by_cases hx : x ∈ Function.support f
· simpa only [coe_add, Pi.add_apply, Function.mem_support, ne_eq, not_not,
Set.piecewise_eq_of_mem _ _ _ hx, _root_.add_left_inj, add_eq_left]
using Set.disjoint_left.1 hff' hx
· simpa only [SimpleFunc.coe_add, Pi.add_apply, Function.mem_support, ne_eq, not_not,
Set.piecewise_eq_of_notMem _ _ _ hx, _root_.add_left_inj, add_eq_right] using hx
rw [this]
exact Measurable.piecewise f.measurableSet_support hf hf'
{g : α → E} (hg : Measurable g) (f : SimpleFunc α E) : Measurable ((f : α → E) + g) :=
f.measurable_bind (fun b a ↦ b + g a) fun b ↦ hg.const_add b

end SimpleFunc

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