example of a partially ordered vector space

Question

Are there any well known examples of partially ordered vector spaces (Riesz Spaces) from applied or pure mathematics?

Answer 1

• The space $l^p$ of all $p$-integrable (resp. bounded if $p = \infty$ sequences is a Riesz space for $1 \le p \le \infty$.
• The space $C([a,b])$ of all continuous functions on the interval $[a,b]$ is a Riesz space.
• Any $L^p(\Omega,\mathcal{A},\mu)$ space [with a measure space $(\Omega,\mathcal{A},\mu)$] is a Riesz space for $1 \le p \le \infty$.
• $C_b(X)$ the space of all continuous bounded real functions on a topological space $X$ is a Riesz space.
• The set of all measurable functions $m(\Omega,\mathcal{A},\mu)$ is a Riesz space. (Modulo nullsets, as always.)
• The set of all $M_\sigma(\Omega,\mathcal{A})$ of all finite signed $\sigma$-additive measures is a Riesz space.
• Similar the $M_{\rm{add}}(\Omega,\mathcal{A})$ of all finitely additive measures is also a Riesz space.