module — Mathlib

English

If μ is finite, every simple function is integrable.

Русский

Если мера конечна, любая простая функция интегрируема.

LaTeX

$$$[IsFiniteMeasure(\\mu)]\\Rightarrow \\mathrm{Integrable}(f,\\mu)$$$

Lean4

/-- If `E` is a normed space, `Lp.simpleFunc E p μ` is a module. Not declared as an
instance as it is (as of writing) used only in the construction of the Bochner integral. -/
protected def module : Module 𝕜 (Lp.simpleFunc E p μ)
    where
  one_smul f := by ext1; exact one_smul _ _
  mul_smul x y f := by ext1; exact mul_smul _ _ _
  smul_add x f g := by ext1; exact smul_add _ _ _
  smul_zero x := by ext1; exact smul_zero _
  add_smul x y f := by ext1; exact add_smul _ _ _
  zero_smul f := by ext1; exact zero_smul _ _

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