I would like to know if the Lebesgue spaces $L_p$ with $ 0 < p < 1 $ are separable or not.
I know that this is true for $1 \leq p < + \infty$, but I do not find any references for the
case $ 0 < p < 1 $.
Thank you
I would like to know if the Lebesgue spaces $L_p$ with $ 0 < p < 1 $ are separable or not.
I know that this is true for $1 \leq p < + \infty$, but I do not find any references for the
case $ 0 < p < 1 $.
Thank you
As Chandru1 stated tentatively, the same arguments apply for $0< p<1$ as for $1\leq p<\infty$, so it is no surprise that a separate proof is hard to find. For example, an introduction to $L^p$ spaces on subsets of $\mathbb{R}^n$ can be found in Chapter 8 of Measure and integral by Wheeden and Zygmund. A proof of separability is outlined for $1\leq p<\infty$ in Theorem 8.15. Part of Theorem 8.16 asserts separability for $0< p<1$, but for proof they simply refer to the proof of 8.15.
Please refer this article. It talks about $L_{p}$ spaces for $0 < p \leq 1$. Link: http://www.jstor.org/stable/2041603?seq=2
Look at the step functions, the ones that take rational values and whose steps have rational endpoints there should be only countably many of those. And then you can perhaps apply the same argument, you use to prove it for $L_{p}$ spaces for $1 < p < \infty$.