integral_norm_condKernel — Mathlib

English

If f is integrable with respect to ρ, then the iterated integral equals the integral of f w.r.t. ρ.

Русский

Если f интегрируема по мере ρ, то двойной интеграл равен интегралу f по мере ρ.

LaTeX

$$$\\int b\\, \\int \\omega\\, f(b,\\omega) \\, \\partial(\\rho.condKernel b) \\, \\partial\\rho.fst = \\int x\\, f(x) \\, \\partial\\rho$$$

Lean4

theorem integral_norm_condKernel {f : α × Ω → F} (hf_int : Integrable f ρ) :
    Integrable (fun x ↦ ∫ y, ‖f (x, y)‖ ∂ρ.condKernel x) ρ.fst :=
  by
  have hf_ae : AEStronglyMeasurable f ρ := hf_int.1
  rw [← hf_ae.ae_integrable_condKernel_iff] at hf_int
  exact hf_int.2

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