PMF integrable of support finite
PMF.integrable_of_support_finite
Plain-language statement
Everything is integrable over PMFs with finite support
Exact Lean statement
lemma PMF.integrable_of_support_finite (f : PMF α) (u : α → ℝ) (hf : f.support.Finite)
[MeasurableSpace α] [MeasurableSingletonClass α] :
MeasureTheory.Integrable u f.toMeasureFormal artifact
Lean source
lemma PMF.integrable_of_support_finite (f : PMF α) (u : α → ℝ) (hf : f.support.Finite) [MeasurableSpace α] [MeasurableSingletonClass α] : MeasureTheory.Integrable u f.toMeasure := by rw [←PMF.restrict_toMeasure_support, ←MeasureTheory.IntegrableOn] have e : f.support = ⋃ (x : α) (_ : x ∈ hf.toFinset), {x} := by ext x simp only [mem_support_iff, ne_eq, Finite.mem_toFinset, mem_iUnion, mem_singleton_iff, exists_prop, exists_eq_right'] rw [e, MeasureTheory.integrableOn_finset_iUnion] intro _ _ rw [MeasureTheory.integrableOn_singleton_iff, PMF.toMeasure_apply_singleton] · right; apply f.apply_lt_top · apply MeasurableSet.singleton- Project
- debate
- License
- Apache-2.0
- Commit
- de3a6e500ae1
- Source
- Prob/Pmf.lean:136-148
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