measure_Iio — Mathlib

English

If f tends to a limit l at +∞, then the measure of Ici(x) is infinite: μ_f(Ici(x)) = ∞.

Русский

Если f стремится к пределу l при +∞, то мера Ici(x) бесконечна: μ_f(Ici(x)) = ∞.

LaTeX

$$μ_f(Ici(x)) = ∞$$

Lean4

theorem measure_Iio {l : ℝ} (hf : Tendsto f atBot (𝓝 l)) (x : ℝ) : f.measure (Iio x) = ofReal (leftLim f x - l) := by
  rw [← Iic_diff_right, measure_diff _ (nullMeasurableSet_singleton x), measure_singleton, f.measure_Iic hf, ←
      ofReal_sub _ (sub_nonneg.mpr <| Monotone.leftLim_le f.mono' le_rfl)] <;>
    simp

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