Measure of countable unions and intersections
Basic properties [ edit ] Let � be a measure. Monotonicity [ edit ] If � 1 and � 2 are measurable sets with � 1 ⊆ � 2 then � ( � 1 ) ≤ � ( � 2 ) . Measure of countable unions and intersections [ edit ] Subadditivity [ edit ] For any countable sequence � 1 , � 2 , � 3 , … of (not necessarily disjoint) measurable sets � � in Σ : � ( ⋃ � = 1 ∞ � � ) ≤ ∑ � = 1 ∞ � ( � � ) . Continuity from below [ edit ] If � 1 , � 2 , � 3 , … are measurable sets that are increasing (meaning that � 1 ⊆ � 2 ⊆ � 3 ⊆ … ) then the union of the sets � � is measurable and � ( ⋃ � = 1 ∞ � � ) = lim � → ∞ � ( � � ) = sup � ≥ 1 � ( � � ) . Continuity from above [ edit ] If � 1 , � 2 , � 3 , … are measurable sets that are decreasing (meaning that � 1 ⊇ � 2 ⊇ � 3 ⊇ … ) then the intersection of the sets � � is measurable; fur...