Theorems · Theorem · measure theory

MeasureTheory.log_rnDeriv_tilted_left_self

∀ {α : Type u_1} {mα : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ} [MeasureTheory.SigmaFinite μ],
  MeasureTheory.Integrable (fun x => Real.exp (f x)) μ →
    (fun x => Real.log ((μ.tilted f).rnDeriv μ x).toReal) =ᵐ[μ] fun x => f x - Real.log (∫ (x : α), Real.exp (f x) ∂μ)

Defined in

Mathlib.MeasureTheory.Measure.Tilted

Cited by

0 results in Mathlib

Foundations

Depth 258 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

MeasureTheory.SigmaFinite