Difference between $P(f(x,w)>0)→1$ at any $x$ and $P(\inf(f(x,w))>0)\to1$ when dimension grows

Question

Let $T:=[-1,1]^{n-1}\times (0,1]$. Let

$$f_n(x_1,\cdots,x_n,w_1,\cdots,w_n):=g(x_1,w_1)+\cdots+g(x_n,w_n)=\sum_{i=1}^ng(x_i,w_i),$$ where

(i) $w_1,\cdots,w_n$ are i.i.d. Gaussian random variables

(ii) $g(x_i,w_i)$ is a smooth function ($g\in C^\infty$)

(iii) $\mathbb{E}g(x_i,w_i)>0$

(iv) $\inf_{(x_1,\cdots,x_n)\in T}\mathbb{E}(\sum_{i=1}^ng(x_i,w_i))=0$

My question: is there any difference between the following two?

(1) for any $(x_1,\cdots,x_n)\in T$, we have $$P(\sum_{i=1}^ng(x_i,w_i)>0)\rightarrow 1$$ when $n$ goes to infinity.

(2) $$P\big(\inf_{(x_1,\cdots,x_n)\in T}\sum_{i=1}^ng(x_i,w_i)>0\big)\rightarrow 1$$ when $n$ goes to infinity.

Thanks for any suggestion!

Answer 1

A trivial counterexample: $$g(x,w):=x.$$

Then convergence (1) holds but convergence (2) does not.

On the other hand, of course, convergence (2) always implies convergence (1).