An inequality for probabilities

Abstract

1. The main result and applications. Given a probability measure space (Q, %, P), let AkEiS, k = l, ■ ■ ' . -V The main result is given in the following theorem. Theorem 1.1. / » 8($>2 (18)Gs2 (1.1) P(U44)^—-■-+ —\*=! / 2a + (2 0)<B 2a + (1 6)<R where ffi = JJml P(Ak), a = YLi Zf-i1-?^* ^ A,) and B = 2a/(& ~[2a/<&], Og0<l. The proof of Theorem 1.1 is given in §2. Corollary 1. A necessary condition for P( \Jt=1Ak) <1 is that (&<(l-t-(l + 8ay'2)/2. Proof. It is easy to verify that if the right-hand side of inequality (1.1) is regarded as a function of 8, then the minimum occurs for 0 = 0. Hence