The first natural thing that come to mind would be to consider the variation of the difference of the measure, i.e. assuming that the limit $$ \lim_{R\to\infty}|\nu-\mu|(B_R) $$ exists.

Considering the desired situation as stated in a comment (i.e. Hausdorff measures with disjoint but close supports), let's rephrase this a bit: The total variation of a measure is indeed a metric on the space of measures, so the above can also be written as $$ \lim_{R\to\infty}d_{TV}(\nu\llcorner B_R,\mu\llcorner B_R) $$ i.e. as the total variation distance of the measures restricted to the balls. As this metric may be too strong for the applications in mind one could replace the metric $d_{TV}$ by weaker metrics and use the condition that $$ \lim_{R\to\infty}d(\nu\llcorner B_R,\mu\llcorner B_R) $$ exists. There are several weaker metrics for measures available, e.g. the ones mentioned in the answers to this question. As an example, one might consider the Kantorovich-Rubinstein metric which is $$ d_{KR}(\mu,\nu) = \sup\{\int f d(\mu-\nu)\mid \mathrm{Lip}(f)\leq 1,\ \|f\|_\infty\leq 1\} $$ i.e. in this particular case $$ d_{KR}(\mu\llcorner B_r,\nu\llcorner B_R) = \sup\{\int_{B_R} f d(\mu-\nu)\mid \mathrm{Lip}(f)\leq 1,\ \|f\|_\infty\leq 1\}. $$ You may play around with the bounds on the Lipschitz constant and the function $f$ as the former is relevant for the measurement of "geometric nearness" while the latter is relevant for the measurement of the mass mismatch.

The first natural thing that come to mind would be to consider the variation of the difference of the measure, i.e. assuming that the limit $$ \lim_{R\to\infty}|\nu-\mu|(B_R) $$ exists.

Considering the desired situation as stated in a comment (i.e. Hausdorff measures with disjoint but close supports), let's rephrase this a bit: The total variation of a measure is indeed a metric on the space of measures, so the above can also be written as $$ \lim_{R\to\infty}d_{TV}(\nu\llcorner B_R,\mu\llcorner B_R) $$ i.e. as the total variation distance of the measures restricted to the balls. As this metric may be too strong for the applications in mind one could replace the metric $d_{TV}$ by weaker metrics and use the condition that $$ \lim_{R\to\infty}d(\nu\llcorner B_R,\mu\llcorner B_R) $$ exists. There are several weaker metrics for measures available, e.g. the ones mentioned in the answers to this question. As an example, one might consider the Kantorovich-Rubinstein metric which is $$ d_{KR}(\mu,\nu) = \sup\{\int f d(\mu-\nu)\mid \mathrm{Lip}(f)\leq 1,\ \|f\|_\infty\leq 1\} $$ i.e. in this particular case $$ d_{KR}(\mu\llcorner B_r,\nu\llcorner B_R) = \sup\{\int_{B_R} f d(\mu-\nu)\mid \mathrm{Lip}(f)\leq 1,\ \|f\|_\infty\leq 1\}. $$ You may play around with the bounds on the Lipschitz constant and the function $f$ as the former is relevant for the measurement of "geometric nearness" while the latter is relevant for the measurement of the mass mismatch.

The first natural thing that come to mind would be to consider the variation of the difference of the measure, i.e. assuming that the limit $$ \lim_{R\to\infty}|\nu-\mu|(B_R) $$ exists.

Considering the desired situation as stated in a comment (i.e. Hausdorff measures with disjoint but close supports), let's rephrase this a bit: The total variation of a measure is indeed a metric on the space of measures, so the above can also be written as $$ \lim_{R\to\infty}d_{TV}(\nu\llcorner B_R,\mu\llcorner B_R) $$ i.e. as the total variation distance of the measures restricted to the balls. As this metric may be too strong for the applications in mind one could replace the metric $d_{TV}$ by weaker metrics and use the condition that $$ \lim_{R\to\infty}d(\nu\llcorner B_R,\mu\llcorner B_R) $$ exists. The standard suspects for metrics would be Prokhorov metric, Kantorovich-Rubinstein metric or the Wasserstein metrics.

The first natural thing that come to mind would be to consider the variation of the difference of the measure, i.e. assuming that the limit $$ \lim_{R\to\infty}|\nu-\mu|(B_R) $$ exists.

Considering the desired situation as stated in a comment (i.e. Hausdorff measures with disjoint but close supports), let's rephrase this a bit: The total variation of a measure is indeed a metric on the space of measures, so the above can also be written as $$ \lim_{R\to\infty}d_{TV}(\nu\llcorner B_R,\mu\llcorner B_R) $$ i.e. as the total variation distance of the measures restricted to the balls. As this metric may be too strong for the applications in mind one could replace the metric $d_{TV}$ by weaker metrics and use the condition that $$ \lim_{R\to\infty}d(\nu\llcorner B_R,\mu\llcorner B_R) $$ exists. There are several weaker metrics for measures available, e.g. the ones mentioned in the answers to this question. As an example, one might consider the Kantorovich-Rubinstein metric which is $$ d_{KR}(\mu,\nu) = \sup\{\int f d(\mu-\nu)\mid \mathrm{Lip}(f)\leq 1,\ \|f\|_\infty\leq 1\} $$ i.e. in this particular case $$ d_{KR}(\mu\llcorner B_r,\nu\llcorner B_R) = \sup\{\int_{B_R} f d(\mu-\nu)\mid \mathrm{Lip}(f)\leq 1,\ \|f\|_\infty\leq 1\}. $$ You may play around with the bounds on the Lipschitz constant and the function $f$ as the former is relevant for the measurement of "geometric nearness" while the latter is relevant for the measurement of the mass mismatch.