Monday, September 11, 2017

Non-measurable sets and intuition

Here’s an interesting reason to accept the existence of non-measurable sets (and hence of whatever weak version of the Axiom of Choice that it depends on). A basic family of mathematical results in analysis says that most measurable real-valued functions on the real line are “close to” being continuous, i.e., that they can be approximated by continuous functions in some appropriate sense. But it is intuitive to think that there “should” be real-valued functions on the real line that are not close to being continuous—there “should” be functions that are very, very messy. So, intuitively, there should be non-measurable functions, and hence non-measurable sets.

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