Identify the function's input and output.

The function assigns to each pair of positive integers the first integer of the pair.
Let the function be $f$. The input is a pair $(m, n)$ where $m$ and $n$ are positive integers.
The output is the first integer of the pair, so $f(m, n) = m$.

The domain is the set of all possible inputs. The problem states the input is "each pair of positive integers".
A positive integer is an element of $\mathbb{Z}^+ = \{1, 2, 3, \dots\}$.
A pair of positive integers is an element of the Cartesian product $\mathbb{Z}^+ \times \mathbb{Z}^+$.
So, the domain is $\mathbb{Z}^+ \times \mathbb{Z}^+$.

The range is the set of all possible outputs. The output of the function is $m$, where $m$ is the first integer of the pair $(m, n)$.
Since $m$ can be any positive integer, the set of all possible values for $m$ is $\mathbb{Z}^+$.
So, the range is $\mathbb{Z}^+$.

The domain is $\boxed{\mathbb{Z}^+ \times \mathbb{Z}^+}$
The range is $\boxed{\mathbb{Z}^+}$