  In the following we construct spaces of dimension $(n\pm \varepsilon)$ lying
in the neighborhood of $\mathbb{Z}^n, \mathbb{Z}^n$ in the context of the
$(n-\varepsilon)$-expansion. We provide means and criteria to deform the spaces
of integer dimension into this neighborhood. We argue that the field theoretic
models living on these deformed spaces are the continuation of the models
defined on the corresponding integer valued spaces. Furthermore we perform the
continuum limit of subgraphs of $\mathbb{Z}^n$ having non-integer dimension to
the corresponding (fractal) subspaces of $\mathbb{Z}^n$. We make sense of a
fractal volume measure like $d^{(n-\varepsilon)}x$.
