Prove2Me

`BookProof.ChapterG.evolution_conserves_probability` {X : Type*} [MeasurableSpace X] (μ : Measure X) [IsProbabilityMeasure μ] (T : X → X) (hT : Measurable T) : IsProbabilityMeasure

Proved
BookProof.ChapterG.evolution_conserves_probability

by leonardopedro · Oct 8, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterG.

BookProof.ChapterG.evolution_conserves_probability {X : Type*} [MeasurableSpace X] (μ : Measure X) [IsProbabilityMeasure μ] (T : X → X) (hT : Measurable T) : IsProbabilityMeasure (μ.map T)

Formalization note: Lean 4 identifier BookProof.ChapterG.evolution_conserves_probability.

Preamble
-- Generated from ChapterG.lean — theorem BookProof.ChapterG.evolution_conserves_probability
import Mathlib
import Definitions.Def_ChapterG
open BookProof.ChapterG


open scoped ComplexConjugate InnerProductSpace Matrix
open MeasureTheory

variable {G : Type*} [Group G] [MeasurableSpace G]
variable {μG : Measure G} [IsProbabilityMeasure μG] [μG.IsMulLeftInvariant]
variable {X : Type*} [MulAction G X]
variable {A : Type*} [Ring A]
Formal statement
theorem BookProof.ChapterG.evolution_conserves_probability {X : Type*} [MeasurableSpace X]
    (μ : Measure X) [IsProbabilityMeasure μ] (T : X → X) (hT : Measurable T) :
    IsProbabilityMeasure (μ.map T) := by sorry

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