• Journal of the Korean Statistical Society
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• v.23 no.2
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• pp.367-378
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• 1994

Let P and Q be probability measures of a measurable space $(\Omega, F)$, and ${F_n}_{n \geq 1}$ be a sequence of increasing sub $\sigma$-fields which generates F. For each $n \geq 1$, let $P_n$ and $Q_n$ be the restrictions of P and Q to $F_n$, respectively. Under the assumption that $Q_n \ll P_n$ for every $n \geq 1$, a zero-one condition is derived for P and Q to have the dichotomy, i.e., either $Q \ll P$ or $Q \perp P$. Then using this condition and the Kolmogorov's zero-one law, we give new and simple proofs of the dichotomy theorems for a pair of Gaussian measures and Poisson processes with examples.