Let $(\Omega, \mathcal{F}, \mathbb{P})$ a probability space and $X_n, X, Y_n, Y$ defined on it taking values in $\mathbb{R}$ with $X_n \to X$ and $Y_n \to Y$ almost surely. Let $f_n, f \colon \mathbb{R} \to \mathbb{R}$ measurable functions satisfying $$ f_n(X_n) \to f(X) $$ almost surely. Further, suppose that the conditional distributions convergence almost surely weakly: $$ \mathcal{L}(X_n \mid Y_n) \stackrel{w}{\to} \mathcal{L}(X \mid Y) $$ in $\mathcal{P}(\mathbb{R})$, almost surely.
$\textbf{Question:}$ Does this imply that $$ \mathcal{L}(f_n(X_n) \mid Y_n) \stackrel{w}{\to} \mathcal{L}(f(X) \mid Y) $$ in $\mathcal{P}(\mathbb{R})$, almost surely?
$\textbf{Ansatz:}$ By the a generalised version of the continuous mapping theorem: If $f_n(x_n) \to f(x)$, whenever $x_n \to x$ (continuous convergence), this holds. However, we have only the weaker assumption that $f_n(X_n) \to f(X)$ almost surely.