lintegral_join_le — Mathlib

English

For f, the lintegral over join m is bounded by the lintegral of the inner lintegral.

Русский

Линтеперел по join ограничен линтепералом внутри.

LaTeX

$$$\\lintegral f \\partial join m \\le \\lintegral (\\lambda μ. \\lintegral f \\partial μ) \\partial m$$$

Lean4

theorem lintegral_join_le (f : α → ℝ≥0∞) (m : Measure (Measure α)) : ∫⁻ x, f x ∂join m ≤ ∫⁻ μ, ∫⁻ x, f x ∂μ ∂m :=
  by
  rcases exists_measurable_le_lintegral_eq (join m) f with ⟨g, hgm, hgf, hfg_int⟩
  rw [hfg_int, lintegral_join hgm.aemeasurable]
  gcongr
  apply hgf

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