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We find upper and lower bounds for the probability of a union of events which generalize the well-known Chung-Erdős inequality. Moreover, we will show monotonicity of the bounds.

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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 may be 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 parts of the Borel-Cantelli lemma.

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