Comment on 245B, notes 3: L^p spaces by Anonymous

It seems that most analysis textbooks, in most cases, only consider the case where the exponent p defining the L^p space satisfies 1 ≤p < ∞ , including these notes here. My question is-what additional structures or properties do L^p spaces with 1 ≤ p < ∞ possess compared to those with 0 ＜ p < 1 that make analysts prefer this "convention"?

Also, even within the range 1 ≤p < ∞ , the cases p = 1, 2, ∞ seem to be the most common, while cases like p = 3, 4, 5, and so forth are relatively rare-Why is that?