Let $(\Omega, \Sigma, \mu)$ be a probability space.
Let $L_p (\mu)$ with fixed $p \geq 1$ be defined to be the collection of all ($\mu$- equivalence classes of) Borel measurable functions $f \colon \Omega \to \mathbb{R}$ for which $\int_{\Omega} |f|^{p} \mathrm{d} \mu < \infty$. Denote by $L_p (\mu)_+$ the positive cone of $L_p (\mu)$.
We say that a subset $E$ of an ordered Banach space (or a Banach lattice) is solid if $E$ contains at least one interior point.
When I read some textbook, it states that "the positive cone $L_p(\mu)_+$ is not solid". Therefore, I'm struggling to figure out how to verify this statement and is this statement always true for any $p \geq 1$? Could anyone help me out please?
If it is convenient to set $\Omega = \mathbb{R}$, please feel free to do it then.
Sincerely appreciate your comment or idea in advance!
Let assume $\Omega = N$,sequence of natural number, then we are dealing with ${l^p}$. Let $x=(x_n)_{n=1}^{\infty}$ with $x_n > 0 $ for all $n$ is an interior point. And take $\lambda >0$ such that $x+y \in l^P_+$ for all $|| y || = \lambda$. There is such a positive $\lambda$ since $x$ is an interior point of $l^p_+$. Let $n$ is large enough such that $x_n < \lambda$. Now define $a= -\lambda e_n$ then $||a|| = \lambda$ but $x+a \notin l^P_+$ since its n-th component is negative! which is a contradiction .