mul_log_le_klDiv — Mathlib

English

A simplified auxiliary relation about toReal KL bounds and ENNReal values used in simplifications.

Русский

Упрощённое вспомогательное соотношение для ограничений toReal KL и ENNReal, применяемых в упрощениях.

LaTeX

$$$\\text{auxiliary lemma: } \\text{(technical relation on toReal and top values)}$$$

Lean4

theorem mul_log_le_klDiv (μ ν : Measure α) [IsFiniteMeasure μ] [IsFiniteMeasure ν] :
    ENNReal.ofReal (μ.real univ * log (μ.real univ / ν.real univ) + ν.real univ - μ.real univ) ≤ klDiv μ ν :=
  by
  by_cases hμν : μ ≪ ν
  swap; · simp [hμν]
  by_cases h_int : Integrable (llr μ ν) μ
  swap; · simp [h_int]
  rw [← ENNReal.ofReal_toReal (a := klDiv μ ν)]
  · exact ENNReal.ofReal_le_ofReal (mul_log_le_toReal_klDiv hμν h_int)
  · rw [klDiv_ne_top_iff]
    exact ⟨hμν, h_int⟩

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