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."

"Uh-huh."

"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."

"Uh-huh."

"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!"