Embedding $L^p$ into weak $L^p$ and notion of boundary of $L^p$ I'm very much a novice to analysis, so this is a kind of conceptual question that is probably very much the wrong way to think about these spaces, but perhaps it fits my more topological brain.
The question is motivated by the following result(s), which can be found in Folland. Fix a measure space $X$ and suppose that $f \in L^{p, \infty}(X)$, and $\mu( \{x: f(x) \neq 0 \} ) < \infty$. Then $f \in L^q(X)$ for all $q < p$. 
We have a flip-side result too, which is that if $f \in L^{p, \infty}(X) \cap L^\infty(X)$, then $f \in L^q$ for all $q > p$.
The Chebychev inequality immediately implies that function that are $L^p$ are also $L^{p,\infty}(X)$, so it seems that those functions that are genuinely weak $L^p(X)$ are like 'edge' cases, functions just barely too bad to not be in $L^p(X)$. The result seems to confirm this, giving all $q$ either just up to or larger than $p$, depending. 
I would like this to manifest itself some way topologically, but this seems suspicious because $L^p(X)$ is already a complete space. So then this leads me to consider the inclusion $L^p(X) \hookrightarrow L^{p, \infty}(X)$. Is the image of $L^p(X)$ dense in some natural topology on $L^{p, \infty}$? What can we say about the embedding map?
 A: I will discuss the case where the measure space is a probability space. For $p\gt 1$, a natural topology for $L^{p,\infty}(X)$ is the one generated by the norm 
$$
\left\lVert Y\right\rVert_{p,\infty}:=\sup\left\{\mu(A)^{1/p-1}\int_A\left\lvert Y\right\rvert\mathrm d\mu,A\in\mathcal F,\mu(A)\gt 0   \right\}.
$$
One can show that the closure of the collection of bounded random variables in $L^{p,\infty}$ endowed with this norm is the collection of random variables 
$Y$ such that $\lim_{k\to +\infty}2^k\mu\left\{\left\lvert Y\right\rvert>2^k\right\}=0$. The closure of $L^p$ is the same. Indeed, using $\left\lVert Y\right\rVert_{p,\infty}\leqslant \left\lVert Y\right\rVert_{p}$ we have for all random variable $Y$ in the closure of $L^p$ for $\left\lVert \cdot\right\rVert_{p,\infty}$ that for each positive $\varepsilon$ there exists a random variables $Z$ in $L^p$ such that $\left\lVert Y-Z\right\rVert_{p,\infty}\lt \varepsilon$. Fix $R$ such that $\lVert Z-Z\mathbf 1\{\left\lvert Z\right\rvert\leqslant R\}\rVert_p\lt \varepsilon$. Then 
$$
\left\lVert Y-Z\mathbf 1\{\left\lvert Z\right\rvert\leqslant R\}\right\rVert_{p,\infty}\leqslant 
\left\lVert Y-Z\right\rVert_{p,\infty}+
\left\lVert  Z-Z\mathbf 1\{\left\lvert Z\right\rvert\leqslant R\}\ \right\rVert_{p,\infty}
\leqslant\varepsilon+\left\lVert  Z-Z\mathbf 1\{\left\lvert Z\right\rvert\leqslant R\}\ \right\rVert_{p}\leqslant 2\varepsilon.
$$ 
