Razborov flag algebras as algebras of measurable functions

These are some brief notes on the translation from Razborov’s recently-developed notion of flag algebra ([9]) into the lexicon of functions and measures on certain abstract Cantor spaces (totally disconnected compact metric spaces). 1 The objects of interest Consider a universal first-order theory T with equality in a language L that contains only predicate symbols; assume T has infinite models. Examples include the theories of undirected and directed graphs and hypergraphs, possibly with loops. In [9], Razborov develops a formalism for handling the ‘leading order’ statistics of large finite models of such theories. The central objects of his theory are the positive Rhomomorphisms of ‘flag algebras’. Here we shall relate these to measures on a subset of the Cantor space ∏ i≤k K N i i (where each Ki is itself some Cantor space) which is compact and such that both set and measure are invariant under the canonical coordinate-permuting action of Sym0(N). Examples of such Cantor spaces include the spaces of models over N of certain kinds of theory, and our identification will begin here. In particular we will identify (under some restrictions) a flag algebra with an algebra of measurable functions on the underlying Cantor space of models, and this will lead to an identification of the positive homomorphisms of the flag algebra with certain measures by virtue of classical results of functional analysis. This will take us through the first three sections of [9]. Section 4, 5 and 6 of [9] relate to a more precise variational analysis of certain examples of these homomorphisms, and we will not discuss this here. For certain special examples of the theories appearing in the study of flag algebras (particularly the theories of hypergraphs and directed hypergraphs),