norm_integral_le_integral_norm — Mathlib

English

For ν a measure, and f integrable with respect to μ+ν, the integral with respect to μ+ν equals the sum of integrals with respect to μ and ν.

Русский

Для меры ν и функции f интегрируемой по μ+ν, интеграл по μ+ν равен сумме интегралов по μ и ν.

LaTeX

$$$$ f_{integral}(μ+ν) = f_{integral}(μ) + f_{integral}(ν) $$$$

Lean4

theorem norm_integral_le_integral_norm (f : α →ₛ E) (hf : Integrable f μ) : ‖f.integral μ‖ ≤ (f.map norm).integral μ :=
  by
  refine (norm_setToSimpleFunc_le_integral_norm _ (fun s _ _ => ?_) hf).trans (one_mul _).le
  exact (norm_weightedSMul_le s).trans (one_mul _).symm.le

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