To specify the probability space is rarely a good idea but since you insist on it, a possible choice would be $\Omega=K^\mathbb N$, where $K=\{1,2,\ldots,k\}$ denotes the set of types of coupons, endowed with the sigma-algebra $\mathcal G=2^K$ and the discrete measure $p=(p(i))_{i\in K}$ on $(K,\mathcal G)$ describing the probability to get a coupon of a given type at each draw. Thus, for every $A\subseteq K$, $A\in\mathcal G$ and $p(A)=\sum\limits_{i\in A}p(i)$.

This considers that one collects coupons again and again, even when one is really interested in what happens with the $m$ first coupons collected.

As a product space, $\Omega$ is endowed with the product sigma-algebra $\mathcal F=\mathcal G^{\otimes\mathbb N}$, generated by the cylinders $A\times K^\mathbb N$ with $A\subseteq K^n$ and $n\in\mathbb N$, and with the product probability measure $\mu=p^{\otimes\mathbb N}$. Thus, $\mu$ is the unique probability measure such that, for every positive $n$ and every $\eta=(\eta_t)_{1\leqslant t\leqslant n}$ in $K^n$, $\mu(\Omega_\eta)=\prod\limits_{t=1}^np(\eta_t)$, where $\Omega_\eta=\{\omega=(\omega_n)_{n\in\mathbb N}\in\Omega\mid\forall t\leqslant n,\omega_t=\eta_t\}$.

For every $\omega=(\omega_n)_{n\in\mathbb N}$ in $\Omega$ and every $n\in\mathbb N$, define $N_n(\omega)$ as the size of the set $\{\omega_t\mid t\leqslant n\}$. Then every $N_n$ is a random variable on $(\Omega,\mathcal F)$. Furthermore, for every positive $\ell$, the number $T_\ell$ of draws needed to collect exactly $\ell$ different coupons is such that $[T_\ell\leqslant m]=[N_m\geqslant\ell]$ and, for every positive $m$, the number $L_m$ of different coupons collected during the $m$ first draws is $L_m=N_m$. This proves that $[T_\ell\leqslant m]=[L_m\geqslant\ell]$ for every positive $m$ and $\ell$.