Show L^p(E) is separable for any measurable E.

[jpriori]

I have a sense that the countable, dense subset I'm looking for is the step functions, maybe over intervals with rational endpoints, but I'm not sure how to deal with the fact that E is any L-msb set, so there's no guarantee all the intervals are in there.

 

[micromass]

You could find an embedding from $L^p(E)$ into $L^p(\mathbb{R})$.

 

[Bacle2]

I have a sense that the countable, dense subset I'm looking for is the step functions, maybe over intervals with rational endpoints, but I'm not sure how to deal with the fact that E is any L-msb set, so there's no guarantee all the intervals are in there.

Don't you mean the simple functions?

 

[Robert1986]

Though it isn't true if p = infinity, is it?

 

[Bacle2]

Though it isn't true if p = infinity, is it?

You're right, L^oo is not separable.

Edit: I think there is an argument using simple functions with rational coefficients. It should
be in most graduate Analysis books.