One of the most common sources of misunderstanding in probability theory is the confusion of an abstract probability distribution with these representations. In an understandable desire to present concrete examples as early as possible, many introductory treatments of probability theory begin immediately with these representations only to end up muddling their relationship with the abstract mathematics that they are characterizing. Now that we have worked through the abstract definitions, however, we can introduce representations in a more principled way.
I’d be more sympathetic to the above if I hadn’t just spent several hours working through what came above that paragraph and understanding basically nothing in each section until I googled the topic at hand and found a relevant example. As it is, I’m just really annoyed.