Theorems · Theorem · measure theory

MeasureTheory.integral_llr_tilted_left

∀ {α : Type u_1} {mα : MeasurableSpace α} {μ ν : MeasureTheory.Measure α} {f : α → ℝ}
  [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.SigmaFinite ν],
  μ.AbsolutelyContinuous ν →
    MeasureTheory.Integrable f μ →
      MeasureTheory.Integrable (MeasureTheory.llr μ ν) μ →
        MeasureTheory.Integrable (fun x => Real.exp (f x)) μ →
          AEMeasurable f ν →
            ∫ (x : α), MeasureTheory.llr (μ.tilted f) ν x ∂μ =
              ∫ (x : α), MeasureTheory.llr μ ν x ∂μ + ∫ (x : α), f x ∂μ - Real.log (∫ (x : α), Real.exp (f x) ∂μ)

Defined in

Mathlib.MeasureTheory.Measure.LogLikelihoodRatio

Cited by

0 results in Mathlib

Foundations

Depth 259 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

MeasureTheory.IsProbabilityMeasureMeasureTheory.SigmaFinite