Documentation

Mathlib.Probability.Kernel.Invariance

Invariance of measures along a kernel #

We say that a measure μ is invariant with respect to a kernel κ if its push-forward along the kernel μ.bind κ is the same measure.

Main definitions #

Push-forward of measures along a kernel #

source
@[deprecated MeasureTheory.Measure.comp_add (since := "2025-02-28")]
theorem ProbabilityTheory.Kernel.bind_add {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} (μ ν : MeasureTheory.Measure α) (κ : Kernel α β) :
(μ + ν).bind ⇑κ = μ.bind ⇑κ + ν.bind ⇑κ
source
@[deprecated MeasureTheory.Measure.comp_smul (since := "2025-02-28")]
theorem ProbabilityTheory.Kernel.bind_smul {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} (κ : Kernel α β) (μ : MeasureTheory.Measure α) (r : ENNReal) :
(r • μ).bind ⇑κ = r • μ.bind ⇑κ
source
@[deprecated ProbabilityTheory.Kernel.comp_const (since := "2025-08-06")]
theorem ProbabilityTheory.Kernel.const_bind_eq_comp_const {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} (κ : Kernel α β) (μ : MeasureTheory.Measure α) :
const α (μ.bind ⇑κ) = κ.comp (const α μ)
source
@[deprecated ProbabilityTheory.Kernel.comp_const (since := "2025-08-06")]
theorem ProbabilityTheory.Kernel.comp_const_apply_eq_bind {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} (κ : Kernel α β) (μ : MeasureTheory.Measure α) (a : α) :
(κ.comp (const α μ)) a = μ.bind ⇑κ

Invariant measures of kernels #

source
def ProbabilityTheory.Kernel.Invariant {α : Type u_1} {mα : MeasurableSpace α} (κ : Kernel α α) (μ : MeasureTheory.Measure α) :
Prop

A measure μ is invariant with respect to the kernel κ if the push-forward measure of μ along κ equals μ.

Equations
Instances For
    source
    theorem ProbabilityTheory.Kernel.Invariant.def {α : Type u_1} {mα : MeasurableSpace α} {κ : Kernel α α} {μ : MeasureTheory.Measure α} (hκ : κ.Invariant μ) :
    μ.bind ⇑κ = μ
    source
    theorem ProbabilityTheory.Kernel.Invariant.comp_const {α : Type u_1} {mα : MeasurableSpace α} {κ : Kernel α α} {μ : MeasureTheory.Measure α} (hκ : κ.Invariant μ) :
    κ.comp (const α μ) = const α μ
    source
    theorem ProbabilityTheory.Kernel.Invariant.comp {α : Type u_1} {mα : MeasurableSpace α} {κ η : Kernel α α} {μ : MeasureTheory.Measure α} (hκ : κ.Invariant μ) (hη : η.Invariant μ) :
    (κ.comp η).Invariant μ