essSup_comp_le_essSup_map_measure

English

For the constant top function, essInf equals top: essInf (const top) μ = top.

Русский

Для константы верх, essInf равен верхней границе: essInf (константа ⊤) μ = ⊤.

LaTeX

$$$$ \\operatorname{essInf} (\\lambda x. \\top) \\mu = \\top $$$$

Lean4

theorem essSup_comp_le_essSup_map_measure (hf : AEMeasurable f μ) : essSup (g ∘ f) μ ≤ essSup g (Measure.map f μ) :=
  by
  refine limsSup_le_limsSup_of_le ?_
  rw [← Filter.map_map]
  exact Filter.map_mono (Measure.tendsto_ae_map hf)

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