TITLE: Structure of an intersection of $L^p$-spaces
QUESTION [9 upvotes]: In what follows, $L^p$ denotes the space of functions from $\mathbb{R}$ to $\mathbb{R}$ such that $\int_{\mathbb{R}} |f(x)|^p\mathrm{d}x < \infty$.
I am interested to understand the structure we can put on intersection of $L^p$-spaces. 
For $I$ an interval of $[1,+\infty]$, we define $$L^I = \bigcap_{p\in I} L^p.$$
Obviously, because $L^a \cap L^b \subset L^c$ for $1\leq a\leq c\leq b \leq \infty$, we have that $L^{[a,b]} = L^a \cap L^b$. Thus, $L^I$ is a Banach space for the norm $\lVert \cdot \lVert_a + \lVert \cdot \lVert_b$.
Questions: What can we say of the structure of $L^I$ if $I$ is an open (or semi-open) interval of $[1,\infty]$? What is the more natural topology on it that make it complete? Are these spaces studied somewhere, for instance as a subpart of a more general theory?
Extensions:

I am also interested in spaces of the form $$L^p_+ = \bigcup_{\epsilon >0}  \bigcap_{0<r<\epsilon} L^{r+p},$$ with similar questions as previously. 
What happens if $I \subset (0,\infty]$, knowing that $L^p$-spaces for $p<1$ are quasi-Banach spaces?

REPLY [2 votes]: I want only to mention you that the paper
Castillo, Díaz & Motos, On the structure of the Fréchet spaces $L_{p-}$, Manuscripta Math. 1998
could be of interest in this regard.