Consider standard Borel spaces $\mathcal{X}, \mathcal{Y}$ and $[0,1]$ with random variables $X, Y, U$ and joint distribution $\mathbb{P}(X, Y, U)$, where the marginal $\mathbb{P}(U) = \lambda$, i.e., $U$ has a uniform distribution on $[0,1]$. Also, there exists a deterministic function $f:\mathcal{X}\times [0,1]\to \mathcal{Y}$ such that $Y = f(X, U)$.
Assume that for conditional distribution $\mathbb{P}(Y|x)$ the mapping $x\mapsto \mathbb{P}(Y|x)$ is continuous (from the topological space $\mathcal{X}$ to the space $\mathcal{P}(Y)$ of probability measures on $Y$, equipped with the weak topology).
So in particular, $x_n \to x$ implies $\mathbb{P}(Y|x_n) \overset{w}{\to} \mathbb{P}(Y|x)$, or equivalently, $\mathbb{P}(f(x_n, U)) \overset{w}{\to} \mathbb{P}(f(x, U))$.
Can we somehow infer that the mapping $x\mapsto f(x, u)$ is $\lambda$-a.e. continuous? Or equivalently, can we show that $f(x_n, U) \overset{a.s.}{\to} f(x, U)$?
