Let $f:\mathbb{R}^p\rightarrow\mathbb{R}^q$ $(p,q\geq 1)$ be a continuous function and $(X_n)_{n\geq 1}$ a sequence of random values on $\mathbb{R}^p$ such that $f(X_n)$ converges in law to a random value $Y$ on $\mathbb{R}^q$. Does there exist a probability space on which $(X_n)_{n\geq 1}, Y$ are defined and $f(X_n)\longrightarrow Y$ almost surely?

Let $f:\mathbb{R}^p\rightarrow\mathbb{R}^q$ $(p,q\geq 1)$ be a continuous function and $(X_n)_{n\geq 1}$ a sequence of random values on $\mathbb{R}^p$ such that $f(X_n)$ converges in law to a random value $Y$ on $\mathbb{R}^q$.

Question: Does there exist a probability space and random values $X_n',Y'$ on it having respectively the same distribution as $X_n$ and $Y$, and such that $f(X_n')\longrightarrow Y'$ a.s?