Is there is any equivalent definitions for double integral, for example in single variable integrals there is these equivalent integrability statements :
$f$ is Riemann integrable $\iff$ for each $\epsilon \gt 0$ , there is a partition $P$ that upholds : $U(P,f) - L(P,f) \lt \epsilon \iff$ for each $\epsilon \gt 0$, there is $\delta \gt 0$, that for each Partition with its norm $\lt \delta$, it upholds that: $U(P,f) - L(P,f) \lt \epsilon$
You can try a similar definition of a multiple Riemann integral.
Caution: it is not necessarily the same as an iterated Riemann integral. Thus for
$$ J = \int_a^b \int_c^d f(x,y)\; dx \; dy$$ it is possible that for each $y \in [a,b]$, the function $x \mapsto f(x,y)$ is Riemann integrable on $[c,d]$, and the resulting Riemann integral $\int_c^d f(x,y)\; dx$ is a Riemann integrable function on $[a,b]$, but the double Riemann integral does not exist.
For example, let $a = c = 0$, $b = d = 2 \pi$, and $$f(x,y) = \sin(n(y) x)$$ where $n(y) = 1$ if $y$ is rational and $0$ if $y$ is irrational. Thus $\int_0^{2\pi} f(x,y)\; dy = 0$ for all $x$. But this does not behave well with respect to partitions.