# What is the meaning of the support set in game theory?

What is the meaning of the support set in game theory? I have seen it, in many papers, however none there explains how did they find it or why did the define it in a specific way. I understand that the support of the mixed strategies contains all the pure strategies that are chosen with positive probability under $$\sigma^*$$, which denotes the probability distribution of the mixed strategies. In other words $$\sigma^*$$ serves also as a probability measure. Can we find the support of the set of the pure strategies as well?

I would appreciate it if someone could give an example apart from an intuitive answer.

Suppose that $$f : X \to \mathbb{R}$$ is a real-valued function whose domain is an arbitrary set $$X.$$ The set-theoretic support of $$f$$, written $$\operatorname{supp}(f),$$ is the set of points in $$X$$ where $$f$$ is non-zero: $$\operatorname{supp}(f) = \{ x \in X \,:\, f(x) \neq 0\}.$$
As you write, in game theory a mixed strategy $$\sigma$$ can be considered a probability measure over pure strategies. That is, it is a function that assigns a number (probability) to all pure strategies, hence $$\operatorname{supp}(\sigma)$$ is the set of pure strategies played with non-zero probability.
An example: in the equilibrium of Prisoner's Dilemma both players play Defect with probability 1, hence the support set of either player's strategy is $$\{ \text{Defect} \}$$.