Theorems · Theorem · information theory

InformationTheory.toReal_klDiv

∀ {α : Type u_1} {mα : MeasurableSpace α} {μ ν : MeasureTheory.Measure α} [MeasureTheory.IsFiniteMeasure μ]
  [MeasureTheory.IsFiniteMeasure ν],
  μ.AbsolutelyContinuous ν →
    MeasureTheory.Integrable (MeasureTheory.llr μ ν) μ →
      (InformationTheory.klDiv μ ν).toReal = ∫ (a : α), MeasureTheory.llr μ ν a ∂μ + ν.real Set.univ - μ.real Set.univ

Defined in: Mathlib.InformationTheory.KullbackLeibler.Basic
Cited by: 3 results in Mathlib
Foundations: Depth 268 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: MeasureTheory.IsFiniteMeasure MeasureTheory.IsFiniteMeasure

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Cites16

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

Realstatement and proof · cited by 25,697
MeasurableSpacestatement and proof · cited by 13,106
MeasureTheory.Measurestatement and proof · cited by 10,939
ENNRealproof · cited by 9,879
Set.univstatement and proof · cited by 3,945
MeasureTheory.integralstatement and proof · cited by 1,779
MeasureTheory.Integrablestatement and proof · cited by 1,367
MeasureTheory.IsFiniteMeasurestatement and proof · cited by 1,078
ENNReal.toRealstatement and proof · cited by 859
MeasureTheory.Measure.realstatement and proof · cited by 530
MeasureTheory.Measure.AbsolutelyContinuousstatement and proof · cited by 325
ENNReal.toReal_ofRealproof · cited by 82

Cited by3

Results whose statement or proof uses this declaration.

InformationTheory.toReal_klDiv_smul_right_eq_smul_leftproof · cited by 3
InformationTheory.toReal_klDiv_smul_leftproof · cited by 1
InformationTheory.toReal_klDiv_of_measure_eqproof · cited by 0