Measurability of scalar products #
theorem
Measurable.inner
{α : Type u_1}
{𝕜 : Type u_2}
{E : Type u_3}
[RCLike 𝕜]
[NormedAddCommGroup E]
[InnerProductSpace 𝕜 E]
{x✝ : MeasurableSpace α}
[MeasurableSpace E]
[OpensMeasurableSpace E]
[SecondCountableTopology E]
{f g : α → E}
(hf : Measurable f)
(hg : Measurable g)
:
Measurable fun (t : α) => Inner.inner 𝕜 (f t) (g t)
theorem
Measurable.const_inner
{α : Type u_1}
{𝕜 : Type u_2}
{E : Type u_3}
[RCLike 𝕜]
[NormedAddCommGroup E]
[InnerProductSpace 𝕜 E]
{x✝ : MeasurableSpace α}
[MeasurableSpace E]
[OpensMeasurableSpace E]
[SecondCountableTopology E]
{c : E}
{f : α → E}
(hf : Measurable f)
:
Measurable fun (t : α) => Inner.inner 𝕜 c (f t)
theorem
Measurable.inner_const
{α : Type u_1}
{𝕜 : Type u_2}
{E : Type u_3}
[RCLike 𝕜]
[NormedAddCommGroup E]
[InnerProductSpace 𝕜 E]
{x✝ : MeasurableSpace α}
[MeasurableSpace E]
[OpensMeasurableSpace E]
[SecondCountableTopology E]
{c : E}
{f : α → E}
(hf : Measurable f)
:
Measurable fun (t : α) => Inner.inner 𝕜 (f t) c
theorem
AEMeasurable.inner
{α : Type u_1}
{𝕜 : Type u_2}
{E : Type u_3}
[RCLike 𝕜]
[NormedAddCommGroup E]
[InnerProductSpace 𝕜 E]
{m : MeasurableSpace α}
[MeasurableSpace E]
[OpensMeasurableSpace E]
[SecondCountableTopology E]
{μ : MeasureTheory.Measure α}
{f g : α → E}
(hf : AEMeasurable f μ)
(hg : AEMeasurable g μ)
:
AEMeasurable (fun (x : α) => Inner.inner 𝕜 (f x) (g x)) μ
theorem
AEMeasurable.const_inner
{α : Type u_1}
{𝕜 : Type u_2}
{E : Type u_3}
[RCLike 𝕜]
[NormedAddCommGroup E]
[InnerProductSpace 𝕜 E]
{m : MeasurableSpace α}
[MeasurableSpace E]
[OpensMeasurableSpace E]
[SecondCountableTopology E]
{μ : MeasureTheory.Measure α}
{f : α → E}
{c : E}
(hf : AEMeasurable f μ)
:
AEMeasurable (fun (x : α) => Inner.inner 𝕜 c (f x)) μ
theorem
AEMeasurable.inner_const
{α : Type u_1}
{𝕜 : Type u_2}
{E : Type u_3}
[RCLike 𝕜]
[NormedAddCommGroup E]
[InnerProductSpace 𝕜 E]
{m : MeasurableSpace α}
[MeasurableSpace E]
[OpensMeasurableSpace E]
[SecondCountableTopology E]
{μ : MeasureTheory.Measure α}
{f : α → E}
{c : E}
(hf : AEMeasurable f μ)
:
AEMeasurable (fun (x : α) => Inner.inner 𝕜 (f x) c) μ