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!"

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