OAI/NumberTheory/PrimePackets/FutureConditionalEntropy.lean

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3 declarations, 0 contain sorry, 0 declared as axioms

Metric entropy from genuine future conditional expectations

  1. lemma OAI.PrimeDegreePackets.futureObservationSigma_monotone line 18
  2. theorem OAI.PrimeDegreePackets.observationEntropyRate_eq_futureConditional line 26
  3. theorem OAI.PrimeDegreePackets.measureEntropy_eq_iSup_futureConditional line 39

    Source documentation

    The metric entropy is computed from actual conditional expectations on the future sigma algebras of explicit finite generating observations.

    theorem measureEntropy_eq_iSup_futureConditional
    [MeasurableSpace.CountablyGenerated Y]
    (μ : Measure Y) [IsProbabilityMeasure μ] {T : Y → Y}
    (hT : MeasurePreserving T μ μ) :
    measureEntropy μ T = ⨆ k, ENNReal.ofReal
      (conditionalSigmaEntropy μ (finiteGeneratingObservation Y k)
        (⨆ n, observationSigma (futureName T (finiteGeneratingObservation Y k) n))) := by
…

    Mentions: OAI.PrimeDegreePackets.measureEntropy, OAI.PrimeDegreePackets.conditionalSigmaEntropy, OAI.PrimeDegreePackets.finiteGeneratingObservation, OAI.PrimeDegreePackets.observationSigma, OAI.PrimeDegreePackets.futureName