eq_iSup_nat — Mathlib

English

For a measure μ of measures, (join map join μ) equals (join (join μ)).

Русский

Для меры мер: join map join μ = join (join μ).

LaTeX

$$$\\mathrm{join}(\\mathrm{map}(\\mathrm{join})\\, \\mu) = \\mathrm{join}(\\mathrm{join}\\, \\mu)$$$

Lean4

theorem eq_iSup_nat (m : Set X → ℝ≥0∞) : mkMetric' m = ⨆ n : ℕ, mkMetric'.pre m n⁻¹ :=
  by
  ext1 s
  rw [iSup_apply]
  refine
    tendsto_nhds_unique (mkMetric'.tendsto_pre_nat m s)
      (tendsto_atTop_iSup fun k l hkl => mkMetric'.mono_pre_nat m hkl s)

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