I am confused about the following statement from Wikipedia:
"A mixed strategy is an assignment of a probability to each pure strategy. This allows for a player to randomly select a pure strategy. Since probabilities are continuous, there are infinitely many mixed strategies available to a player."
How do we check the Nash equilibrium inequality then?
For example, in Bayesian games with player set $N=\{1,2,\ldots,n\}$, the equilibrium is defined as the strategy profile $s=(s_i,s_{-i})$ such that, we have $$EU_i(s_i,s_{-i})\geq EU_i(s_i',s_{-i}),$$ for all $s_i'\in S_i$, for all $i\in N$ and $s_i'\neq s_i$.
In many texts, $S_i$ is defined as the function $S_i:\Theta_i\to\Pi(A_i)$, where $\Theta_i$ is the type set and $\Pi(A_i)$ is the probability distribution over the actions.
How is it possible to have infinitely many mixed strategies? What is $S_i$ exactly, a set or a function?