  Comparing probability distributions is an indispensable and ubiquitous task
in machine learning and statistics. The most common way to compare a pair of
Borel probability measures is to compute a metric between them, and by far the
most widely used notions of metric are the Wasserstein metric and the total
variation metric. The next most common way is to compute a divergence between
them, and in this case almost every known divergences such as those of
Kullback--Leibler, Jensen--Shannon, R\'enyi, and many more, are special cases
of the $f$-divergence. Nevertheless these metrics and divergences may only be
computed, in fact, are only defined, when the pair of probability measures are
on spaces of the same dimension. How would one quantify, say, a KL-divergence
between the uniform distribution on the interval $[-1,1]$ and a Gaussian
distribution on $\mathbb{R}^3$? We show that these common notions of metrics
and divergences give rise to natural distances between Borel probability
measures defined on spaces of different dimensions, e.g., one on $\mathbb{R}^m$
and another on $\mathbb{R}^n$ where $m, n$ are distinct, so as to give a
meaningful answer to the previous question.
