Wassermans "All of Nonparametric Statistics" from Springer defines a finite sample confidence set as: $$ \inf_{F\in\mathcal{F}}P(\theta\in C_n)\geq1-\alpha $$ Where $\theta$ is a parameter of interest, $C_n$ is a set of possible values of $\theta$, and $F$ is a distribution among a set of possible ones called a statistical model $\mathcal{F}$. I'm fairly new to the $\inf$ operator and don't know how to interpret it. I only know it as the greatest lower bound aka kind of like the "lowest value" of this confidence interval but what does a lowest F mean? Or is it not talking about F (I assume so because it's under the operator)? I'm interpreting this to be like the best F such that we can get a probability of our $C_n$ containing $\theta$ higher than our desired confidence level, but I'm missing the final step as to what this inf is meaning.
And I have even less hope of understanding a uniform asymptotic confidence set: $$ \liminf_{n\to\infty} \inf_{F\in\mathcal{F}} P(\theta\in C_n) \geq 1-\alpha $$
When taking the $\inf_{F\in \mathcal{F}}$, you have assume that F is some distribution in $\mathcal{F}$. Once you have assumed that, now $P(\theta\in C_n)\geq1-\alpha$, will be a curve or function (in a way) and you have to find the minimum(infimum) of that. So, the role $F\in \mathcal{F}$ is just to tell you that there is an F of such kind and you have to apply infimum operator on such distribution $F$. For this curve or function that I'm talking about depends on different arguments, $\theta \, ,n \, and \, \,\alpha$. What $\liminf_{n\to\infty}$ does is that it tells you that one of those arguments n now tends to infinity and now you have to find infimum assuming that.