Process of Understanding Sigma-Algebra

It turns out looking out for a statistical insight in my everyday life is quite difficult, and I think part of the reason why is that I am too distracted by my Stochastic Processes assignment to think about statistics outside of the class. It has been only a week since I started to write Statistics Diary, and I already run out of examples of statistics I find in my life. So today, I am going to write about concepts that I'm mystified with. In the first few classes of Stochastic Processes, we learned the actual definitions of outcomes, events, sample space, sigma-algebra, and measure—all of which I “thought” I knew, but apparently was not. I am especially mystified the concept of sigma-algebra: I get there is the empty set, and sample space omega. And there’s subsets of the omega, A’s. Sigma-algebra contains all the possible kinds of A’s, but what I don’t get is how sigma-algebra is conceptually different from omega—aren't they both the bigger set that contains all the subsets??

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As I wrote the last sentence, I could finally picture how sigma-algebra is different from omega—omega may not necessarily contain subset A’s complement, but sigma-algebra must contain it! I’m really glad I got this, but I do feel silly about laying this whole thought process down.