| import Mathlib | |
| namespace OAI | |
| open MeasureTheory | |
| open scoped NNReal ENNReal | |
| lemma probability_three_complements {Ω : Type*} [MeasurableSpace Ω] | |
| (μ : Measure Ω) [IsProbabilityMeasure μ] (S T U : Set Ω) | |
| (hS : MeasurableSet S) (hT : MeasurableSet T) (hU : MeasurableSet U) | |
| (hμS : μ S ≤ (1/64:ENNReal)) (hμT : μ T ≤ (1/64:ENNReal)) | |
| (hμU : μ U ≤ (1/64:ENNReal)) : | |
| (1/2:ENNReal) ≤ μ ((S ∪ T ∪ U)ᶜ) := by | |
| have hb : μ (S ∪ T ∪ U) ≤ (1/64:ENNReal)+(1/64:ENNReal)+(1/64:ENNReal) := | |
| (measure_union_le _ _).trans (add_le_add ((measure_union_le _ _).trans (add_le_add hμS hμT)) hμU) | |
| rw [prob_compl_eq_one_sub ((hS.union hT).union hU)] | |
| refine le_trans ?_ (tsub_le_tsub_left hb 1) | |
| apply (ENNReal.toReal_le_toReal (by finiteness) (by finiteness)).mp | |
| rw [ENNReal.toReal_sub_of_le (by | |
| apply (ENNReal.toReal_le_toReal (by finiteness) (by finiteness)).mp | |
| norm_num [ENNReal.toReal_add]) (by finiteness)] | |
| norm_num [ENNReal.toReal_add] | |
| end OAI | |