jpriori said: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.
Robert1986 said:Though it isn't true if p = infinity, is it?
L^p(E) is a function space that contains all measurable functions on a measurable space E that satisfy a certain norm condition. This space is commonly used in the study of functional analysis and measure theory.
A function space is considered separable if it contains a countable dense subset. In other words, there exists a sequence of functions that can approximate any function in the space with arbitrary precision.
Showing that L^p(E) is separable is important because it allows for the use of certain approximation techniques and simplifies mathematical proofs. It also helps in the study of various properties of the space, such as the existence of solutions to certain equations.
There are several approaches to proving the separability of L^p(E). One method is to construct a countable set of simple functions that are dense in the space. Another method is to use the Rademacher functions, which are known to be dense in L^p(E) for any measurable E.
Yes, the separability of L^p(E) holds for all values of p. This is because the techniques used to prove separability are independent of the value of p and only rely on the properties of measurable functions on a measurable space E.