  Let $R=\mathbb{F}_p[x_1,\ldots,x_n]$ and let $\mathbf{F}$ be the ring of
Frobenius operators over $R$. We introduce a notion of Bernstein dimension and
multiplicity for the class of finitely generated $\mathbf{F}$-modules whose
structure morphism has a finite length kernel. We show that an
$\mathbf{F}$-module belongs to this class if and only if it admits a great
filtration with respect to the Bernstein filtration on $\mathbf{F}$. We
describe the Hilbert series of these great filtrations, and prove that the
dimension and multiplicity defined in terms of this Hilbert series are
independent of the choice of filtration. We refer to the $\mathbf{F}$-modules
of Bernstein dimension $0$ as $F$-holonomic. We show that $F$-holonomic
$\mathbf{F}$-modules are a full abelian subcategory of
$\text{Mod}_{\mathbf{F}}$, closed under taking extensions, on which
multiplicity is an additive function. We show that Lyubeznik's finitely
generated unit $\mathbf{F}$-modules are $F$-holonomic.
