Converging pointwise almost-surely versus almost-surely pointwise

Question

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?

Answer 1

In a) there is a null set for each $y$. b) insists that there is a null set independent of $y$. The two are not equivalent. For example, let $X$ have some continuous distribution, say normal, and $X_n=X$ for all $n$. Let $g(x,y)=1$ if $x=y$, and $g(x,y)=0$ if $x\neq y$. Let $h \equiv 0$. The you can check that a) holds, but the probability in b) is $0$, not $1$!