Theorems · Theorem · measure theory

essSup_map_measure

∀ {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α}
  [inst : ConditionallyCompleteLattice β] {γ : Type u_3} {mγ : MeasurableSpace γ} {f : α → γ} {g : γ → β}
  [inst_1 : MeasurableSpace β] [inst_2 : TopologicalSpace β] [SecondCountableTopology β] [OrderClosedTopology β]
  [OpensMeasurableSpace β],
  AEMeasurable g (MeasureTheory.Measure.map f μ) →
    AEMeasurable f μ →
      autoParam (Filter.IsCoboundedUnder (fun x1 x2 => x1 ≤ x2) (MeasureTheory.ae (MeasureTheory.Measure.map f μ)) g)
          essSup_map_measure._auto_1 →
        autoParam (Filter.IsBoundedUnder (fun x1 x2 => x1 ≤ x2) (MeasureTheory.ae μ) (g ∘ f))
            essSup_map_measure._auto_3 →
          autoParam (Filter.IsCoboundedUnder (fun x1 x2 => x1 ≤ x2) (MeasureTheory.ae μ) (g ∘ f))
              essSup_map_measure._auto_5 →
            autoParam
                (Filter.IsBoundedUnder (fun x1 x2 => x1 ≤ x2) (MeasureTheory.ae (MeasureTheory.Measure.map f μ)) g)
                essSup_map_measure._auto_7 →
              essSup g (MeasureTheory.Measure.map f μ) = essSup (g ∘ f) μ

Defined in

Mathlib.MeasureTheory.Function.EssSup

Cited by

3 results in Mathlib

Foundations

Depth 204 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

ConditionallyCompleteLatticeMeasurableSpaceTopologicalSpaceSecondCountableTopologyOrderClosedTopologyOpensMeasurableSpace