When this definition came up in the (one and only) measure theory course I took as a student, the instructor (Peter Constantin) had this to say about it:

"It says that a set is measurable if you can make change with it."

It says something that this explanation has stuck in my mind for the last 15 years, but it is possible that I have remembered it in part because I was never really sure I understood what he meant. Anyway, it sounds good, and if I ever teach a measure theory course (I shudder to imagine the apocalyptic scenario that would necessitate me to be called upon to do this...) I might pass it along to my students.

When this definition came up in the (one and only) measure theory course I took as a student, the instructor (Peter Constantin) had this to say about it:

"It says that a set is measurable if you can make change with it."

This explanation has stuck in my mind for the last 15 years, but it is possible that I have remembered it in part because I was never really sure I understood what he meant. Anyway, it sounds good, and if I ever teach a measure theory course (I shudder to imagine the apocalyptic scenario that would necessitate my being called upon to do this: will there be any other mathematicians at all? what color will the sky be?) I might pass it along to my students.