integral_zero_measure — Mathlib

English

If the measure is the zero measure, the Bochner integral of any function is zero: ∫ f d0 = 0.

Русский

Если мера равна нулевой мере, интеграл Боchnerа любой функции равен нулю: ∫ f d0 = 0.

LaTeX

$$$$\int x, f(x) \partial(0) = 0.$$$$

Lean4

@[simp]
theorem integral_zero_measure {m : MeasurableSpace α} (f : α → G) : (∫ x, f x ∂(0 : Measure α)) = 0 :=
  by
  by_cases hG : CompleteSpace G
  · simp only [integral, hG, L1.integral]
    exact setToFun_measure_zero (dominatedFinMeasAdditive_weightedSMul _) rfl
  · simp [integral, hG]

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