Measurability of real and complex functions #
We show that most standard real and complex functions are measurable, notably exp
, cos
, sin
,
cosh
, sinh
, log
, pow
, arcsin
, arccos
.
See also MeasureTheory.Function.SpecialFunctions.Arctan
and
MeasureTheory.Function.SpecialFunctions.Inner
, which have been split off to minimize imports.
theorem
Real.measurable_of_measurable_exp
{α : Type u_1}
:
∀ {x : MeasurableSpace α} {f : α → ℝ}, (Measurable fun (x : α) => (f x).exp) → Measurable f
theorem
Real.aemeasurable_of_aemeasurable_exp
{α : Type u_1}
:
∀ {x : MeasurableSpace α} {f : α → ℝ} {μ : MeasureTheory.Measure α},
AEMeasurable (fun (x : α) => (f x).exp) μ → AEMeasurable f μ
theorem
Measurable.exp
{α : Type u_1}
{m : MeasurableSpace α}
{f : α → ℝ}
(hf : Measurable f)
:
Measurable fun (x : α) => (f x).exp
theorem
Measurable.log
{α : Type u_1}
{m : MeasurableSpace α}
{f : α → ℝ}
(hf : Measurable f)
:
Measurable fun (x : α) => (f x).log
theorem
Measurable.cos
{α : Type u_1}
{m : MeasurableSpace α}
{f : α → ℝ}
(hf : Measurable f)
:
Measurable fun (x : α) => (f x).cos
theorem
Measurable.sin
{α : Type u_1}
{m : MeasurableSpace α}
{f : α → ℝ}
(hf : Measurable f)
:
Measurable fun (x : α) => (f x).sin
theorem
Measurable.cosh
{α : Type u_1}
{m : MeasurableSpace α}
{f : α → ℝ}
(hf : Measurable f)
:
Measurable fun (x : α) => (f x).cosh
theorem
Measurable.sinh
{α : Type u_1}
{m : MeasurableSpace α}
{f : α → ℝ}
(hf : Measurable f)
:
Measurable fun (x : α) => (f x).sinh
theorem
Measurable.sqrt
{α : Type u_1}
{m : MeasurableSpace α}
{f : α → ℝ}
(hf : Measurable f)
:
Measurable fun (x : α) => √(f x)
theorem
Measurable.cexp
{α : Type u_1}
{m : MeasurableSpace α}
{f : α → ℂ}
(hf : Measurable f)
:
Measurable fun (x : α) => (f x).exp
theorem
Measurable.ccos
{α : Type u_1}
{m : MeasurableSpace α}
{f : α → ℂ}
(hf : Measurable f)
:
Measurable fun (x : α) => (f x).cos
theorem
Measurable.csin
{α : Type u_1}
{m : MeasurableSpace α}
{f : α → ℂ}
(hf : Measurable f)
:
Measurable fun (x : α) => (f x).sin
theorem
Measurable.ccosh
{α : Type u_1}
{m : MeasurableSpace α}
{f : α → ℂ}
(hf : Measurable f)
:
Measurable fun (x : α) => (f x).cosh
theorem
Measurable.csinh
{α : Type u_1}
{m : MeasurableSpace α}
{f : α → ℂ}
(hf : Measurable f)
:
Measurable fun (x : α) => (f x).sinh
theorem
Measurable.carg
{α : Type u_1}
{m : MeasurableSpace α}
{f : α → ℂ}
(hf : Measurable f)
:
Measurable fun (x : α) => (f x).arg
theorem
Measurable.clog
{α : Type u_1}
{m : MeasurableSpace α}
{f : α → ℂ}
(hf : Measurable f)
:
Measurable fun (x : α) => (f x).log