Kurzbeschreibung: The mathematical definition of probability distributions for random functions, measures or generalized functions is not straightforward. Analogous to Kolmogorov’s theorem for the existence of random functions, we consider the existence of limits of inverse systems of random histograms (obtained as projections of measures onto their restrictions to finite partitions). We find that such limits exist as random (bounded/positive/probability) measures, that manifest in four distinct phases distinguished by their concentration properties. Applications include a careful analysis of the existence proof for the Dirichlet and Polya tree processes from Bayesian statistics/machine learning, as well as the well-known Gaussian Free Field (GFF) from probability/quantum field theory. The phase of the GFF depends on the dimension: in one (time-like) dimension, the GFF describes a signed measure with Brownian motion as its random density function; in (space-time) dimensions greater than one, the GFF is in a phase that describes a random -generalized- density function and in which `particles' emerge as the natural (Poisson-process type) manifestation of the random inverse limit measure.