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f205175 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 | import Mathlib
import OAI.Probability.CriticalPercolation.ProductMeasure.FiniteMarkovAvg
namespace OAI
section
section
open Set MeasureTheory ProbabilityTheory
open Set
open scoped BigOperators
namespace CriticalPercolation.FiniteMarkov
variable {X : Type*} [Fintype X]
lemma avg_div (μ : X → ℝ) (a : ℝ) (f : X → ℝ) :
avg (fun x => μ x/a) f = avg μ f/a := by
simp only [avg, Finset.sum_div]
apply Finset.sum_congr rfl
intro x _
ring
lemma stationary_of_avg (K : X → X → ℝ) (μ : X → ℝ)
(h : ∀ f : X → ℝ, avg μ (step K f) = avg μ f) (y : X) :
∑ x, μ x * K x y = μ y := by
classical
simpa [step, avg] using h (fun x => if x = y then 1 else 0)
end CriticalPercolation.FiniteMarkov
end
end
end OAI
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