AutoSamplingTheory.TechnicalLemmas.Probability.KernelTransport

import Mathlib.Probability.Kernel.Invariance

/-!
# Invariance under measurable changes of coordinates

Transport a kernel and its invariant measure through a measurable equivalence.
The result applies to arbitrary kernels and measures, without Markovness,
finiteness, s-finiteness, Standard Borel or nonempty assumptions.
-/

namespace AutoSamplingTheory.TechnicalLemmas.Probability.KernelTransport

open MeasureTheory ProbabilityTheory

/-- Conjugating a kernel by a measurable equivalence preserves invariance of
the corresponding pushforward measure. The input is pulled back with `e.symm`
and the output is pushed forward with `e`.

This is exact invariance only, not reversibility or convergence. The equivalence
supplies genuine measurability, so no nonmeasurable-map fallback is used. -/
theorem invariant_map_comap {α β : Type*} [MeasurableSpace α] [MeasurableSpace β]
    {κ : Kernel α α} {μ : Measure α} (e : α ≃ᵐ β) (hκ : κ.Invariant μ) :
    ((κ.comap e.symm e.symm.measurable).map e).Invariant (μ.map e) := by
  change ((κ.comap e.symm e.symm.measurable).map e) ∘ₘ μ.map e = μ.map e
  rw [← Measure.map_comp _ _ e.measurable,
    ← Kernel.comp_deterministic_eq_comap, ← Measure.comp_assoc,
    Measure.deterministic_comp_eq_map, e.map_symm_map, hκ.def]

end AutoSamplingTheory.TechnicalLemmas.Probability.KernelTransport