  It is proven that the physical measure for the two-dimensional Yang-Mills
theory is purely singular with respect to the kinematical Ashtekar-Lewandowski
measure. For this, an explicit decomposition of the gauge orbit space into
supports of these two measures is given. Finally, the results are extended to
more general (e.g. confining) theories. Such a singularity implies, in
particular, that the standard method of determining the physical measure via
``exponential of minus the action times kinematical measure'' is not
applicable.
