Definition (Weak Containment): Let $G$ be a locally compact group, and let $\pi, \rho$ be unitary representations of $G$ into Hilbert spaces $\mathcal{H}$ and $\mathcal{K}$, respectively. Then $\pi$ is weakly contained in $\rho$ if for every $x \in \mathcal{H}$, for every compact set $K \subseteq G$, and for every $\epsilon > 0$, there exist $y_1,y_2,...,y_n \in \mathcal{K}$ such that for all $g \in K$, we have
$$\left| \langle \pi (g)x,x \rangle - \sum_{i=1}^{n} \langle \rho(g)y_i,y_i \rangle \right| < \epsilon$$
I've searched for the definition of containment of unitary representations, but I wasn't able to find it. Does anyone know what it means to say that $\pi$ is contained in $\rho$?
To say that $\pi$ is contained in $\rho$ is the same as saying that $\pi$ is a subrepresentation of $\rho$, i.e., there is an isometry $V:\mathcal H\to\mathcal K$ such that $V\pi(g)V^*=\rho(g)$ for all $g\in G$.