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

1. Apr 23, 2013

### 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.

2. Apr 23, 2013

### micromass

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

3. Apr 25, 2013

### Bacle2

Don't you mean the simple functions?

4. May 5, 2013

### Robert1986

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

5. May 10, 2013

### Bacle2

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.

Last edited: May 10, 2013