Let $X$ be a random variable valued in $F$, $\epsilon>0$ be fixed, $f$ a measurable nonnegative function and $S\subseteq F$ a compact subset. I know that $$\forall x\in S: P(f(X,x)>\epsilon)\leq P(\sup_{x\in S}f(X,x)>\epsilon)$$ due to the monotonicity of $P$, and so, $\sup_{x\in S}P(f(X,x)>\epsilon)\leq P(\sup_{x\in S}f(X,x)>\epsilon)$. My question is when does the equality hold: $$ \sup_{x\in S}P(f(X,x)>\epsilon)= P(\sup_{x\in S}f(X,x)>\epsilon)?$$
Comments Maybe, the R.H.S. involves limit of sets, while the $L.H.S.$ involves limit of real sequences. Then both sides can be related through monotone convergence theorem.
If you want more information, I'm actually working with the random sequence $(Y_i,X_i)$ valued in $\mathbb R\times F$ where $(F,d)$ is a metric space. In my case, $X=(Y_i,X_i)_{i=1}^n$ and $f(X,x)=\sum_{i=1}^n \lvert Y_i\rvert 1_{B(x,h)}(X_i) $ with $B(x,h)=\{x'\in F:d(x',x)\leq h\}$ (a closed ball in $F$).