iIndepSets_iff — Mathlib

English

If f and g are identically distributed, then their e-variance equality holds; in particular, evariance(f, μ) = evariance(g, ν).

Русский

Если f и g identically distributed, то e-variance равна: evariance(f, μ) = evariance(g, ν).

LaTeX

$$$\text{IdentDistrib } f g μ ν \Rightarrow evariance f μ = evariance g ν$$$

Lean4

theorem iIndepSets_iff (π : ι → Set (Set Ω)) (μ : Measure Ω) :
    iIndepSets π μ ↔
      ∀ (s : Finset ι) {f : ι → Set Ω} (_H : ∀ i, i ∈ s → f i ∈ π i), μ (⋂ i ∈ s, f i) = ∏ i ∈ s, μ (f i) :=
  by simp only [iIndepSets, Kernel.iIndepSets, ae_dirac_eq, Filter.eventually_pure, Kernel.const_apply]

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