# Question about Supremum/Infimum over double indexed Sets

I admittedly have a fairly stupid question about a problem, which I wanted to make sure I understand correctly, since I actually never came across this before. Let $$f$$ be a function, $$f: \mathbb{R^2} \rightarrow \mathbb{R}$$. What does $$\inf_{x \in \mathbb{R}} {\sup_{y\in \mathbb{R}}}f(x,y)$$ exactly mean? If i want to evaluate the supremum first, do I look at every $$x$$ but treat it as a constant value or do I just pick one x and evaluate $$f(x,y_1),...,f(x,y_n),...$$ and then take the smallest upper bound of that? And what does the infimum over that then look like? I am kind of confused about that notation, appreciate any help. Thank you in advance!

Edit: My original problem is the following. Let $$g$$ be an estimator for $$\theta$$. The risk is given by $$R(\theta,g)$$. I am now supposed to evaluate $$\inf_{g} \sup_{\theta \in \Theta}R(\theta,g)$$, where $$\Theta$$ is the parameter set and $$g$$ is an estimator. This is used in Stochastics and Probability theory.

You can think of it in the following way. First, define the function $$g\colon \mathbb R \to \mathbb R$$ by $$g(x) = \sup_{y \in \mathbb R} f(x,y).$$ Then, letting $$C = \inf_{x\in\mathbb R} \sup_{y\in\mathbb R} f(x,y)$$, we have that $$C = \inf_{x\in\mathbb R} g(x).$$
In other words, you can find $$C$$ by first finding the supremum over $$y$$ of $$f(x,y)$$ for each $$x$$. This gives you the function $$g$$ described above. Then, you can find the infimum of $$g$$ over $$x$$.
Note that, in general, $$\sup_{y\in\mathbb R} \inf_{x\in\mathbb R} f(x,y) \le C.$$ This means you should not do the process I described above in reverse, as you may sometimes have strict inequality above. (See Wikipedia.) The conditions where you have equality is the subject of a number of different theorems. See, for example, Sion's Theorem.