An example of posets from analysis. If a measure space \\((X,\mu)\\) is \\(\sigma\\)-finite, then its \\(L^p\\) spaces (the collection of measurable functions \\(f\colon X\to\Bbb C\\) such that \\(\int_X|f|^p\,\rm{d}\mu<\infty\\)) can be preordered by set inclusion. In general this is not a total order. However, consider a finite measure space \\((X,\mu)\\). As a consequence of HÃ¶lder's inequality, we can actually put a total order on the \\(L^p\\) spaces with inclusion: $$ L^q(X,\mu)\subseteq L^p(X,\mu)\qquad\text{if $1\le p\le q\le \infty$.} $$

I would be interested if anybody knew of any interesting follow-up points to this post, since while it's true that we can do this, it feels somewhat contrived.

I would be interested if anybody knew of any interesting follow-up points to this post, since while it's true that we can do this, it feels somewhat contrived.