`BookProof.ChapterCompactCompleteReducibility.compact_invariant_complement` {ρ : G →* (V ≃L[ℂ] V)} (hρ : Continuous fun g => (ρ g : V →L[ℂ] V)) (μ : Measure G) [IsProbabilityMeasur

Proved

BookProof.ChapterCompactCompleteReducibility.compact_invariant_complement

by leonardopedro · Oct 7, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterCompactCompleteReducibility.

BookProof.ChapterCompactCompleteReducibility.compact_invariant_complement {ρ : G →* (V ≃L[ℂ] V)} (hρ : Continuous fun g => (ρ g : V →L[ℂ] V)) (μ : Measure G) [IsProbabilityMeasure μ] [μ.IsMulLeftInvariant] (W : Submodule ℂ V) (hW : ∀ g : G, ∀ x ∈ W, ρ g x ∈ W) : ∃ W' : Submodule ℂ V, (∀ g : G, ∀ x ∈ W', ρ g x ∈ W') ∧ IsCompl W W'

Formalization note: Lean 4 identifier BookProof.ChapterCompactCompleteReducibility.compact_invariant_complement.

Preamble

-- Generated from ChapterCompactCompleteReducibility.lean — theorem BookProof.ChapterCompactCompleteReducibility.compact_invariant_complement
import Mathlib
import Definitions.Def_ChapterCompactCompleteReducibility
open BookProof.ChapterCompactCompleteReducibility



open MeasureTheory

variable {G : Type*} [Group G] [TopologicalSpace G] [IsTopologicalGroup G] [CompactSpace G]
  [MeasurableSpace G] [BorelSpace G] {μ : Measure G} [IsProbabilityMeasure μ]
  [μ.IsMulLeftInvariant]
variable {V : Type*} [NormedAddCommGroup V] [NormedSpace ℂ V] [FiniteDimensional ℂ V]

Formal statement

theorem BookProof.ChapterCompactCompleteReducibility.compact_invariant_complement {ρ : G →* (V ≃L[ℂ] V)}
    (hρ : Continuous fun g => (ρ g : V →L[ℂ] V)) (μ : Measure G) [IsProbabilityMeasure μ]
    [μ.IsMulLeftInvariant] (W : Submodule ℂ V) (hW : ∀ g : G, ∀ x ∈ W, ρ g x ∈ W) :
    ∃ W' : Submodule ℂ V, (∀ g : G, ∀ x ∈ W', ρ g x ∈ W') ∧ IsCompl W W' := by sorry

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