  The Reifenberg theorem \cite{reif_orig} tells us that if a set $S\subseteq
B_2\subseteq \mathbb R^n$ is uniformly close on all points and scales to a
$k$-dimensional subspace, then $S$ is H\"older homeomorphic to a
$k$-dimensional Euclidean ball. In general this is sharp, for instance such an
$S$ may have infinite volume, be fractal in nature, and have no rectifiable
structure.
  The goal of this note is to show that we can improve upon this for an almost
calibrated Reifenberg set, or more generally under a positivity condition in
the context of an $\epsilon$-calibration $\Omega$ . An $\epsilon$-calibration
is very general, the condition holds locally for all continuous $k$-forms such
that $\Omega[L]\leq 1+\epsilon$ for all $k$-planes $L$. We say an oriented
$k$-plane $L$ is $\alpha$-positive with respect to $\Omega$ if
$\Omega[L]>\alpha>0$. If $\Omega[L]>\alpha> 1-\epsilon$ then we call $L$ an
$\epsilon$-calibrated plane.
  The main result of this paper is then the following. Assume at all points and
scales $B_r(x)\subseteq B_2$ that $S$ is $\delta$-Hausdorff close to a subspace
$L_{x,r}$ which is uniformly positive $\Omega[L_{x,r}]>\alpha $ with respect to
an $\epsilon$-calibration. Then $S$ is $k$-rectifiable with uniform volume
bounds.
