Let $(\Omega, \mathcal{E}, \mathbb{P})$ be a probability space, and $X_n$ $(n=1,2,\dots)$ a sequence of random variables, and $g: \mathbb{R}^2 \to \mathbb{R}$ a measurable function, and $h: \mathbb{R} \to \mathbb{R}$ a real-valued function.
Are the following two statements different?
(a) $g(y, X_n) \to h(y)$ almost-surely $\mathbb{P}$, pointwise in $y$.
(b) $g(y, X_n) \to h(y)$ pointwise in $y$, almost surely $\mathbb{P}$.
I think that (a) means:
$$ \forall y. \mathbb{P}[\omega \in \Omega \mid g(y, X_n(\omega)) \to h(y)] = 1. $$
whereas (b) means: $$ \mathbb{P}[ \omega \in \Omega \mid \forall y. g(y, X_n(\omega)) \to h(y)] = 1, $$
(assuming that the event in (b) is measurable for simplicity).
Although I'm still not sure what would be the difference between these two expressions, or if this interpretation makes sense. Is one implied by the other?