Dumb/Challenging conventional wisdom question possibly related to my previous question.
Why do we sometimes consider a measure space $(S, \Sigma, \mu) = (\mathbb{R}, \mathscr{B}(\mathbb{R}), \lambda)$ where $\lambda$ is Lebesgue measure rather than $(S, \Sigma, \mu) = (\mathbb{R}, \mathscr{M}(\mathbb{R}), \lambda)$ where $\mathscr{M}(\mathbb{R})$ is the set of $\lambda$-measurable subsets of $\mathbb{R}$? I mean, there are subsets of $\mathbb{R}$ that are not Borel sets but $\lambda$-measurable right? If there are none, I guess that answers the first question.
Possibly answered by above but why, in my previous question, is it 'natural' to consider $\mathscr{F}$? I'm guessing it's like why it's 'natural' to consider $\mathscr{B}(\mathbb{R})$.
Possibly related:
Why do probabilists take random variables to be Borel (and not Lebesgue) measurable?