IsReversible — Mathlib

English

A reversible Markov kernel leaves the invariant measure π unchanged.

Русский

Обратимое марковское ядро оставляет неизменной инвариантную меру π.

LaTeX

$$$IsReversible(\\kappa, \\pi) \\Rightarrow Invariant(\\kappa, \\pi)$$$

Lean4

/-- Reversibility (detailed balance) of a Markov kernel `κ` w.r.t. a measure `π`:
for all measurable sets `A B`, the mass flowing from `A` to `B` equals that from `B` to `A`. -/
def IsReversible (κ : Kernel α α) (π : Measure α) : Prop :=
  ∀ ⦃A B⦄, MeasurableSet A → MeasurableSet B → ∫⁻ x in A, κ x B ∂π = ∫⁻ x in B, κ x A ∂π

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