I have noticed that there are two definitions in Riemann integral(not multiple).
Definition 1. For all $\epsilon > 0$, there exists $\delta > 0$ such that for any tagged partition $x_0$, ..., $x_n$ and $t_0$, ..., $t_{n − 1}$ whose mesh is less than $\delta$, we have $$|\sum_{i=0}^{n-1}f(t_i)(x_i-x_{i-1})-S|<\epsilon$$
Definition 2. For all $\epsilon > 0$, there exists a tagged partition $y_0$, ..., $y_m$ and $r_0$, ..., $r_{m − 1}$ such that for any tagged partition $x_0$, ..., $x_n$ and $t_0$, ..., $t_{n − 1}$ which is a refinement of $y_0$, ..., $y_m$ and $r_0$, ..., $r_{m − 1}$, we have $$|\sum_{i=0}^{n-1}f(t_i)(x_i-x_{i-1})-S|<\epsilon$$ But I also notice that it seems to need a property to prove Definition 1 from Definition 2, which is that the number of the intervals in a partition which "go across" another interval in another partition is finite.But it doesn't seem to have the property still when it goes to multiple integral.
So can we prove the equivalence between the correspondent two definitions in multiple integral just like Riemann integral?That is does multiple integral have the second definition?