The "parameter" in a Bayesian nonparametric problem is the unknown distribution P of the observation X. A Bayesian uses a prior distribution for P, and after observing X, solves the statistical inference problem by using the posterior distribution of P, which is the conditional distribution of P given X. For Bayesian nonparametrics to be successful one needs a large class of priors for which posterior distributions can be easily calculated.Unless X takes values in a finite space, the unknown distribution P varies in an infinite dimensional space. Thus one has to talk about measures in a complicated space like the space of all probability measures on a large space. This has always required a more careful attention to the attendant measure theoretic problems.A class of priors known as Dirichlet measures have been used for the distribution of a random variable X when it takes values in 7?k, see Freedman (1963), Fabius (1964 and Ferguson (1973). This family forms a conjugate family and possesses many pleasant properties.In this paper we give a simple and new constructive definition of Dirichlet measures and remove the restriction that the basic space should be lk.We give complete self contained proofs of the three basic results for Dirichlet measures:1. The Dirichlet measure is a probability measure of on the space of all probability measures, 2. it gives probability one to the subset ot discrete probability measures, and 3. the posterior distribution is also a Dirichlet measure.

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The vast majority of reliability analyses assume that components and system are in either of two states: functioning or failed. The present paper develops basic theory for the study of systems of components in which any of a finite number of states may occur, representing at one extreme perfect functioning and at the other extreme complete failure. We lay down axioms extending the standard notion of a coherent system to the new notion of a multistate coherent system. For such systems we obtain deterministic and probabilistic properties for system performance which are analogous to well-known results for coherent system reliability.

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