Tangent points of lower content $d$-regular sets and $\beta$ numbers

Given a lower content $d$-regular set in $\mathbb{R}^n$, we prove that the subset of points in $E$ where a certain Dini-type condition on the so-called Jones $\beta$ numbers holds coincides with the set of tangent points of $E$, up to a set of $\mathcal{H}^d$-measure zero. The main point of our result is that $\mathcal{H}^d|_E$ is not assumed to be $\sigma$-finite; because of this, we use a certain variant of the $\beta$ coefficient, firstly introduced by Azzam and Schul in [AS1], which is given in terms of integration with respect to the Hausdorff content.