prior measures are used to construct admissible procedures, by appeal to the characterization of admissible procedures as Bayesian procedures (or limits of Bayesian procedures) by Wald. For example, a right Haar measure for a family of distributions with a location parameter results in the Pitman estimator, which is best equivariant. When left and right Haar measures differ, the right measure is usually preferred as a prior distribution. For the group of affine transformations on the parameter space of the normal distribution, the right Haar measure is the Jeffreys prior measure. Unfortunately, even right Haar measures sometimes result in useless priors, which cannot be recommended for practical use, like other methods of constructing prior measures that avoid subjective information. Another use of Haar measure in statistics is in conditional inference, in which the sampling distribution of a statistic is conditioned on another statistic of the data. In invariant-theoretic conditional inference, the sampling distribution is conditioned on an invariant of the group of transformations (with respect to which the Haar measure is defined). The result of conditioning sometimes depends on the order in which invariants are used and on the choice of a maximal invariant, so that by itself a statistical principle of invariance fails to select any unique best conditional statistic (if any exist); at least another principle is needed. For non-compact groups, statisticians have extended Haar-measure results using amenable groups. Weil's converse theorem In 1936, André Weil proved a converse (of sorts) to Haar's theorem, by showing that if a group has a left invariant measure with a certain separating property, then one can define a topology on the group, and the completion of the group is locally compact and the given measure is essentially the same as the Haar measure on this completion. See also Invariant measure Pontryagin duality Riesz–Markov–Kakutani representation theorem Notes Further reading . André Weil, Basic Number Theory, Academic Press, 1971. External links The existence and uniqueness of the Haar integral on a locally compact topological group - by Gert K. Pedersen On the Existence and Uniqueness of Invariant Measures on Locally Compact Groups - by Simon Rubinstein-Salzedo Lie groups Topological groups Measures (measure theory) Harmonic analysis