integral_Iio_add_Ici — Mathlib

English

Another variant of the interval additivity/combinatorial identity.

Русский

Еще одно вариативное тождество аддитивности интервалов.

LaTeX

$$$ \\int_{a}^{b} f(x) \\, dμ - \\int_{c}^{d} f(x) \\, dμ = \\int_{?}^{?} f(x) \\, dμ - \\int_{?}^{?} f(x) \\, dμ $$$

Lean4

theorem integral_Iio_add_Ici (h_left : IntegrableOn f (Iio b) μ) (h_right : IntegrableOn f (Ici b) μ) :
    (∫ x in Iio b, f x ∂μ) + (∫ x in Ici b, f x ∂μ) = ∫ (x : ℝ), f x ∂μ :=
  by
  convert (setIntegral_union (Iio_disjoint_Ici <| Eq.le rfl) measurableSet_Ici h_left h_right).symm
  rw [Iio_union_Ici, Measure.restrict_univ]

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