I was going to write about measurable functions, but after a little discussion about measurable sets with a friend, I decided to put that off for a bit. The discussion had to do with the nature of measurable sets and how weird they could look. He notes (paraphrased!):
"I mean, open sets are measurable. Those are nice. And closed sets are measurable. Those are nice."
"But there’s all sorts of sets which aren’t open or closed which are measurable. I mean, there’s sorts of sets which don’t even look like finite unions of open and closed sets."
"So to say that measurable sets are ‘nice’ in some way is really not an accurate statement. We can make a measurable set as not-nice as we want it!"