snd_integral_withLp — Mathlib

English

If f is integrable on μ and t is a measurable subset, the restricted integral preserves the fst-coordinate relation.

Русский

Если f интегрируема на μ и t — измеримое подмножество, ограниченный интеграл сохраняет fst-координату.

LaTeX

$$$$\\text{Integrable}(f,\\mu) \\Rightarrow \\int x\\, (f x)\\,d(\\mu\\restriction t)_{\\mathrm{fst}} = \\int x\\, (f x)_{\\mathrm{fst}}\\,d(\\mu\\restriction t).$$$$

Lean4

theorem snd_integral_withLp [CompleteSpace E] (hf : Integrable f μ) : (∫ x, f x ∂μ).snd = ∫ x, (f x).snd ∂μ :=
  by
  rw [← WithLp.ofLp_snd]
  conv => enter [1, 1]; change WithLp.prodContinuousLinearEquiv q ℝ E F _
  rw [← ContinuousLinearEquiv.integral_comp_comm, snd_integral]
  · rfl
  · exact (ContinuousLinearEquiv.integrable_comp_iff _).2 hf

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