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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 | import Mathlib
import OAI.Probability.CriticalPercolation.ProductMeasure.Density
namespace OAI
section
section
open Set MeasureTheory ProbabilityTheory
open scoped BigOperators
namespace CriticalPercolation.FiniteProduct
variable {E A : Type*} [Fintype E] [Fintype A]
lemma density_mul_le_expect {w : E → A → ℝ} (hw : ∀ e a, 0 ≤ w e a)
{f : (E → A) → ℝ} (hf : ∀ ω, 0 ≤ f ω) (ω : E → A) :
density w ω * f ω ≤ expect w f := by
classical
exact Finset.single_le_sum (fun ω _ => mul_nonneg (density_nonneg hw ω) (hf ω))
(Finset.mem_univ ω)
end CriticalPercolation.FiniteProduct
end
end
end OAI
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