# Definition of Stable convergence in law: why do we need an extension of the probability space?

I am trying to understand the definition of stable convergence in law. I have found the following definition.

Definition. Let $Y_n$be a sequence of random variables defined on a probability space $(\Omega,\mathcal{F},\mathbb{P})$ with value in $\mathbb{R}^d$. We say that $Y_n$ converges stably with limit $Y$, where $Y$ is defined on an extension $(\Omega^{\prime},\mathcal{F}^{\prime},\mathbb{P}^{\prime})$, iff for any bounded, continuous function $g$ and for any bounded $\mathcal{F}$-measurable random variable $Z$ it holds that

$$\mathbb{E}[g(Y_n)\,Z]\longrightarrow\mathbb{E}^{\prime}[g(Y)\,Z]\text{ as }n\rightarrow\infty$$

My problem is: why do we need an extension of the probability space? And how this extension is generally defined?

Lemma : The following are equivalent $$Y_n\rightarrow Y \quad stably \\ (Y_n,Z)\xrightarrow{d}(Y,Z)$$ for any $\mathcal{F}$-mb. rv. $Z$.
Proposition: Assume $Y_n\rightarrow Y$ stably, where $Y$ is $\mathcal{F}$-mb. Then $Y_{n}\xrightarrow{P} Y$.
Proof: Since $Y_n\rightarrow Y \quad stably$ and $Y$ is $\mathcal{F}$-mb, we know by the Lemma, that $(Y_n,Y)\xrightarrow{d}(Y,Y)$. Thus $Y_n -Y\xrightarrow{d} 0$ which means, that $Y_n\xrightarrow{P} Y$.
However your Definition connects two concepts, the stable convergence, and the weak $L_{1}$-conergence.