In this paper we consider the algebraic crossed product${\mathcal{A}}:=C_{K}(X)\rtimes _{T}\mathbb{Z}$induced by a homeomorphism$T$on the Cantor set$X$, where$K$is an arbitrary field with involution and$C_{K}(X)$denotes the$K$-algebra of locally constant$K$-valued functions on$X$. We investigate the possible Sylvester matrix rank functions that one can construct on${\mathcal{A}}$by means of full ergodic$T$-invariant probability measures$\unicode[STIX]{x1D707}$on$X$. To do so, we present a general construction of an approximating sequence of$\ast$-subalgebras${\mathcal{A}}_{n}$which are embeddable into a (possibly infinite) product of matrix algebras over$K$. This enables us to obtain a specific embedding of the whole$\ast$-algebra${\mathcal{A}}$into${\mathcal{M}}_{K}$, the well-known von Neumann continuous factor over$K$, thus obtaining a Sylvester matrix rank function on${\mathcal{A}}$by restricting the unique one defined on${\mathcal{M}}_{K}$. This process gives a way to obtain a Sylvester matrix rank function on${\mathcal{A}}$, unique with respect to a certain compatibility property concerning the measure$\unicode[STIX]{x1D707}$, namely that the rank of a characteristic function of a clopen subset$U\subseteq X$must equal the measure of$U$.