I learned about sigma algebras in the context of financial mathematics, where the emphasis is on proving things about certain pathologies of probability which obtain in continuous in certain pathological limits of uncountable what-have-yous, in order to demonstrate that you are clever enough to get good grades proving that stuff in order that you can ignore it for the rest of your career as a financial trader trading in discrete time where none of those theorems matter a damn.
However, ignore your Dynkin lemma for a moment, and sit back and consider what sigma algebras gives us even in the non-fiddly Tarski-paradox cases:
A way of classifying the arguments we need to evaluate (measure) functions.
Discuss: faithfulness in graphical models. What else can we generalise this to?
Define measurability-with-respect-to as opposed to measurability. Look at this for non-measure functions, (we do this implicitly all the time in probability with measure-valued functions) and non-set functions (e.g. markings of the sets in the domain).
Also to think about here: the relationship between the much-ignored underlying event-space and the observations. See also copulas, pseudorandomness