integral_eq_integral_of_support_subset

English

A rearrangement identity for sub-interval integrals involving a,b,c,d.

Русский

Перестановочная формула для интегралов по субинтервалам с параметрами a,b,c,d.

LaTeX

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

Lean4

theorem integral_eq_integral_of_support_subset {a b} (h : support f ⊆ Ioc a b) : ∫ x in a..b, f x ∂μ = ∫ x, f x ∂μ :=
  by
  rcases le_total a b with hab | hab
  · rw [integral_of_le hab, ← integral_indicator measurableSet_Ioc, indicator_eq_self.2 h]
  · rw [Ioc_eq_empty hab.not_gt, subset_empty_iff, support_eq_empty_iff] at h
    simp [h]

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