On lower and upper bounds for probabilities of unions and the   Borel--Cantelli lemma
  We obtain new lower and upper bounds for probabilities of unions of events.These bounds are sharp. They are stronger than earlier ones. General bounds maybe applied in arbitrary measurable spaces.We have improved the method that has been introduced in previous papers. We derive new generalizations of the first and second part of the Borel--Cantelli lemma. 
Andrei N. Frolov
math.PR