Theorems · Theorem · real analysis

AbsolutelyContinuousOnInterval.fun_mul

∀ {a b : ℝ} {f g : ℝ → ℝ},
  AbsolutelyContinuousOnInterval f a b →
    AbsolutelyContinuousOnInterval g a b → AbsolutelyContinuousOnInterval (fun i => f i * g i) a b

Eta-expanded form of AbsolutelyContinuousOnInterval.mul If f and g are absolutely continuous on uIcc a b, then f * g is absolutely continuous on uIcc a b.

Defined in: Mathlib.MeasureTheory.Function.AbsolutelyContinuous
Cited by: 1 results in Mathlib
Foundations: Depth 168 from the axioms · uses propext, Classical.choice, Quot.sound

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Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.

Realstatement · cited by 25,697
AbsolutelyContinuousOnIntervalstatement · cited by 31
AbsolutelyContinuousOnInterval.mulproof · cited by 1

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AbsolutelyContinuousOnInterval.integral_deriv_mul_eq_subproof · cited by 1