If our goal is to derive bound on $P(\sup_{x\in T} f(x)\geq a)$, where $T$ is a uncountably infinite subset $[0,1]^n$ of $\mathbb{R}^n$, and $f(x):=\sum_{i=1}^n 1+w_i$ ($w_i$ is gaussian scalar), then we would need epsilon-net argument, given that for any $x$, the bound on $P(f(x)\geq a)\leq\exp(-a^2/2)$. However, I am confused on the difference between
(1) the random variable $P(f(x)\geq a)\leq p$, for any $x$
(2) $P(\sup_{x\in T} f(x)\geq a)=P(\forall x,f(x)\geq a)\leq p$.
$a$ is parameter.
Which one is stronger?
Update:
I think $P(\sup_{x\in T} f(x)\geq a)=P(\forall x,f(x)\geq a)$. But, if we want to derive bound on $P(\sup_{x\in T} f(x)\geq a)$, from any $x$, $P(f(x)\geq a)$, we need epsilon net argument, since union bound here does not work.
Am I correct?