Mathematical notation $\max$ with simple example for non-mathematician

$$\max_{x \in X} f(x)$$ is the notation we use for "the maximum value of $f(x)$ when $x$ is allowed to vary throughout the set $X$".

For example, $$\max_{x \in \{1,2,3\}} x = 3$$ $$\max_{x \in \{1,2,3\}} \frac{5}{x} = 5$$ $$\max_{\theta \in \mathbb{R}} \sin(\theta) = 1$$

The notation $\max_{x \in X}f(x)$ means the number $\max \{ f(x) | x \in X \}$. Likewise, we have $\min_{x \in X}f(x) = \min \{ f(x)| x \in X \}$.

If you want to denote a point $y$ at which $f$ reaches its extremum, use $\arg \max _{x \in X}f(x)$ or $\arg \min_{x \in X}f(x)$ accordingly.

The notation $\min_{t \in T}\max_{x \in X}f_{t\in T}(x)$ simply means the number $\min \{ \max_{x \in X}f_{t}(x) | t \in T \}$; intuitively, first find the maximum for each $t \in T$ and then find the minimum of the maxima of these $f_{t}$.

Note that supremum and infimum are a generalization of maximum and minimum, respectively. If $f$ is continuous and if $X$ is compact, it can be proved that $\sup f(X)$ and $\inf f(X)$ are the functional values of $f$ somewhere in $X$; in this case we have $\sup f(X) = \max f(X)$ and $\inf f(X) = \min f(X)$.

$$\max_{\theta\in\Theta}R_T(\theta)$$means the maximum value that $R_T(\theta)$ can take as $\theta$ varies over all possible values in $\Theta$. This quantity is still dependent on $T$, and its least value for all possible values of $T$ is $$\min_{T}\max_{\theta\in\Theta}R_T(\theta).$$ Here the possible values for $T$ are assumed to be understood, and not made explicit. Also, the meaning is the same if the letter $T$ is replaced by $T_1$ or some other appropriate symbol.