I was looking up Egorov's Theorem on Wikipedia. http://en.wikipedia.org/wiki/Egorov%27s_theorem
One of the conditions is that the functions attain values in a separable metric space M, and in the "Discussion of assumptions" section following the theorem, the following is stated: "The separability of the metric space is needed to make sure that for M-valued, measurable functions f and g, the distance d(f(x), g(x)) is again a measurable real-valued function of x."
Can someone explain how to prove the boldfaced statement?
[EDIT]To be precise, I want to know how to prove "If the metric space M is separable, and the functions f and g are measurable functions attaining values in M, then the distance function d(f(x),g(x)) is measurable."