norm_eq_integral — Mathlib

English

A reprise of the monotonicity property for integrals with respect to dominating measures.

Русский

Повторение свойства монотонности интеграла по доминирующим мерам.

LaTeX

$$$$ \\int f \\le \\int g \\quad \\text{when } f \\le g \\text{ a.e. and integrable}. $$$$

Lean4

theorem norm_eq_integral (f : α →₁ₛ[μ] E) : ‖f‖ = ((toSimpleFunc f).map norm).integral μ :=
  by
  rw [norm_eq_sum_mul f, (toSimpleFunc f).map_integral norm (SimpleFunc.integrable f) norm_zero]
  simp_rw [smul_eq_mul]

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