  We describe a $\tilde{O}(d^{5/6})$-query monotonicity tester for Boolean
functions $f:[n]^d \to \{0,1\}$ on the $n$-hypergrid. This is the first $o(d)$
monotonicity tester with query complexity independent of $n$. Motivated by this
independence of $n$, we initiate the study of monotonicity testing of
measurable Boolean functions $f:\mathbb{R}^d \to \{0,1\}$ over the continuous
domain, where the distance is measured with respect to a product distribution
over $\mathbb{R}^d$. We give a $\tilde{O}(d^{5/6})$-query monotonicity tester
for such functions.
  Our main technical result is a domain reduction theorem for monotonicity. For
any function $f:[n]^d \to \{0,1\}$, let $\epsilon_f$ be its distance to
monotonicity. Consider the restriction $\hat{f}$ of the function on a random
$[k]^d$ sub-hypergrid of the original domain. We show that for $k =
\text{poly}(d/\epsilon)$, the expected distance of the restriction is
$\mathbb{E}[\epsilon_{\hat{f}}] = \Omega(\epsilon_f)$. Previously, such a
result was only known for $d=1$ (Berman-Raskhodnikova-Yaroslavtsev, STOC 2014).
Our result for testing Boolean functions over $[n]^d$ then follows by applying
the $d^{5/6}\cdot \text{poly}(1/\epsilon,\log n, \log d)$-query hypergrid
tester of Black-Chakrabarty-Seshadhri (SODA 2018).
  To obtain the result for testing Boolean functions over $\mathbb{R}^d$, we
use standard measure theoretic tools to reduce monotonicity testing of a
measurable function $f$ to monotonicity testing of a discretized version of $f$
over a hypergrid domain $[N]^d$ for large, but finite, $N$ (that may depend on
$f$). The independence of $N$ in the hypergrid tester is crucial to getting the
final tester over $\mathbb{R}^d$.
