Theorems · Theorem · information theory

InformationTheory.toReal_klDiv_smul_left

∀ {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 →
      ∀ (c : NNReal),
        (InformationTheory.klDiv (c • μ) ν).toReal =
          ↑c * (InformationTheory.klDiv μ ν).toReal + (1 - ↑c) * ν.real Set.univ + ↑c * Real.log ↑c * μ.real Set.univ

Defined in: Mathlib.InformationTheory.KullbackLeibler.Basic
Cited by: 1 results in Mathlib
Foundations: Depth 269 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: MeasureTheory.IsFiniteMeasure MeasureTheory.IsFiniteMeasure

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Cites41

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

DFunLike.coeproof · cited by 62,936
Realstatement and proof · cited by 25,697
MeasurableSpacestatement and proof · cited by 13,106
MeasureTheory.Measurestatement and proof · cited by 10,939
ENNRealstatement · cited by 9,879
NNRealstatement and proof · cited by 4,310
Set.univstatement and proof · cited by 3,945
one_mulproof · cited by 2,841
add_zeroproof · cited by 2,707
Nat.cast_oneproof · cited by 2,501
zero_addproof · cited by 2,366
MeasureTheory.aeproof · cited by 2,352

Cited by1

Results whose statement or proof uses this declaration.

InformationTheory.toReal_klDiv_smul_rightproof · cited by 0