L^p space related question for p= infinity

[smanalysis]

Homework Statement

We are asked to exhibit a measurable set E such that L^p(E) is separable for p= infinity. Also, we have to show that L^infinity(E) is not separable if E contains a nondegenerate interval.

Homework Equations

A normed linear space X is separable provided there is a countable subset that is dense in X.
In my real analysis class, we showed that L^infinity[a,b] is not separable but then my professor said that L^infinity of a countable set will be separable.

The Attempt at a Solution

Going by what my professor said, I'm thinking that L^infinity(Q) would work here, but I'm having a little trouble starting out on how to show that this is separable. Any ideas? I would really appreciate some help here. For the second part of the problem, I think it just follows from the fact that L^infinity[a,b] is not separable, but I'm struggling to figure out why. Well, wouldn't [a,b] itself consist of disjoint sets of nondegenerate intervals? Then again, maybe this question is related to something different...

 

[micromass]

Going by what my professor said, I'm thinking that L^infinity(Q) would work here, but I'm having a little trouble starting out on how to show that this is separable. Any ideas?

OK, you COULD proceed this way, but you're making it hard on yourself. What about L^\infty(\{0\}), so simply take a singleton. Isn't that a lot easier??

I would really appreciate some help here. For the second part of the problem, I think it just follows from the fact that L^infinity[a,b] is not separable, but I'm struggling to figure out why.

Let's fix some notations, let E be our set that contains the nondegenerate interval [a,b]. You could begin by showing that there exists an embedding L^\infty([a,b])\rightarrow L^\infty(E).

Well, wouldn't [a,b] itself consist of disjoint sets of nondegenerate intervals?

Yes, of course, but I don't see how this would help you...

 

[Dick]

I would listen to micromass' advice. Especially since I don't think L^infinity(Q) is separable. Why don't you try to show that it isn't? It's a pretty simple Cantor diagonal type argument.