  We develop a new technique for studying monomial ideals in the standard
polynomial rings $A[X_1,\ldots,X_d]$ where $A$ is a commutative ring with
identity. The main idea is to consider induced ideals in the semigroup ring
$R=A[\mathbb{M}^1_{\geq 0}\times\cdots\times\mathbb{M}^d_{\geq 0}]$ where
$\mathbb{M}^1,\ldots,\mathbb{M}^d$ are non-zero additive subgroups of
$\mathbb{R}$. We prove that the set of non-zero finitely generated monomial
ideals in $R$ has the structure of a metric space, and we prove that a version
of Krull dimension for this setting is lower semicontinuous with respect to
this metric space structure. We also show how to use discrete techniques to
study certain monomial ideals in this context.
