A generalization of the Chung-Erdös inequality for the probability of the union of events

A generalization of the Chung–Erdös inequality for the probability of the union of arbitrary events is proved using some lower bounds for tail probabilities. We present a lower bound for the probability of appearance of at least $m$ events from the set of events $A_1,\dots,A_n$ where $1\le m\le n$.