A certain class of total variation measures of analytic measures

In this paper we investigate a problem concerning the total variation measure of an analytic measure induced by a flow. Our main results are: Let μ be a positive Baire measure on a compact Hausdorff space and let the distant future in L²(μ) be the zero subspace. If μ is absolutely continuous with respect to an invariant measure, then μ is the total variation measure of an analytic measure. On the other hand, if μ is singular with respect to each invariant measure, then there is a summable Baire function g such that gdμ is analytic and g⁻¹ is bounded. Moreover, we note that general μ can be uniquely expressed as the sum of measures of above two types.