The Measure for the Multiverse and the Probability for Inflation

We investigate the measure problem in the framework of inflationary cosmology. The measure of the history space is constructed and applied to inflation models. Using this measure, it is shown that the probability for the generalized single field slow roll inflation to last for $N$ e-folds is suppressed by a factor $\exp(-3N)$, and the probability for the generalized $n$-field slow roll inflation is suppressed by a much larger factor $\exp(-3nN)$. Some non-inflationary models such as the cyclic model do not suffer from this difficulty.Comment: 16 page