I have recently been stuck trying to understand how game theorists extend a normal form game (matrix game) into a game with mixed strategies (so called mixed extension). I feel like I am missing something obvious, perhaps an implicit identification.

Take for example these notes. At the very bottom near the references section, it reads:

Let $\Sigma_i$ denote the set of probability distributions over $S_i$ ($S_i$ is the set of pure strategies of player $i$)., i.e., $\sigma_i \in \Sigma_i$ if and only if $\sigma_i$ is probability measure.

My confusion in that $\sigma_i$ elsewhere in the game literature typically refers to the mixed strategies, which are vectors $\sigma_i \in \Delta$. For example, in a normal form game with three strategies it can be $\sigma_i = [1,0,0]^T$ (which represents a pure strategy) or $\sigma_i = [1/3, 1/3, 1/3]^T$ rather than measurable function. When people refer to mixed strategies (see Simple proof of the existence of Nash equilibria for 2-person games?), they usually refer to these probability vectors. Is there an implicit identification between these measurable functions and these vectors?

Another confusion arises at the definition of the expected utility (right before the references),

With this assumption, the utility functions $u_i$ are extended from $S = \prod_{j = 1}^I S_j$ to the space of probability distributions $\Sigma = \Pi_{j = 1}^I \Sigma_j$ as follows: $$ u_i(\sigma) = \int_C u_i(s) d\sigma(s)$$ where $\sigma \in \Sigma$

Once again, elsewhere in the literature, the expected utility $u_i$ usually is defined to take from the product of the simplex $\Delta = \prod_{i = 1}^I \Delta_i$ to a real number, i.e., $u_i: \Delta \to \mathbb{R}$ rather than $u_i: \Sigma \to \mathbb{R}$ . Here, the expected utility is taking from a product of measurable spaces. Once again, is there an identification that the authors are making?

So my question is: are mixed strategies in game theory measurable functions or elements of the simplex? If the former, then the expected utility is essentially a composition of measurable functions, is that right? Or is there a common but implicit identification between these two employed in the game theory literature? Or more plainly: what is a mixed strategy? I'm new to game theory so sorry in advance if this is a well known thing.

I've dug up some paper that explicitly defines mixed strategies as measurable function (and even pure strategies as functions). But this seems to be rare.

Greatly appreciated if anyone has a good reference on the construction from normal form game to games with mixed strategies or some references on game theory from measure theory point of view.