in this post I gave a definition and some properties for the spaces $\text{gr}^p(\mu)$ that are mentioned here. It is somehow an addendum to my previous question, but it is quite long so I add it in a new post. So here I prove two things:
$1$. Let $X$ be a set and $(x_n)\subset X$. We define the measure $\delta$ on $(X,\mathcal{P}(X))$ as $\delta(A):=\displaystyle{\sum_{i=1}^\infty c_i\delta_i(A)}$ where $c_i\in[0,\infty)$ and $\delta_i$ are the Dirac point-masses of weight $1$ on $x_i$. Then it is $\text{gr}^p(\delta)=c_0$.
Indeed, let $(a_n)\in c_0$. If $g\in L^p(\delta)$ it is $\|g\|_p=\displaystyle{\bigg{(}\sum_{n=1}^\infty c_i|g(x_i)|^p\bigg{)}^{1/p}}$. Now if $(f_n)\subset L^p(\mu)$ with $\|f_n\|_p\leq |a_n|$ for all $n$ then for any $i$ it is $c_i|f_n(x_i)|^p\leq\|f_n\|_p^p\leq|a_n|^p\to 0$. Therefore $|f_n(x_i)|\to0$ for all $i$ and since our measure is $\delta$, this is precisely $f_n\to0$ $\delta$-a.e. on $X$.
$2$. Let $(\mathbb{R}^d,\mathcal{L}^d,\lambda_d)$ be the d-dimensional euclidean space equipped with the Lebesgue measure. Then $\text{gr}^p(\lambda_d)=\ell^p$.
Suppose that there exists $(a_n)\in\text{gr}^p(\lambda_d)-\ell^p$ and we will get a contradiction: since $\displaystyle{\sum_{n=1}^\infty|a_n|^p=\infty}$ we define the following integers:
$m_1:=\min\{n\in\mathbb{N}:|a_1|^p+\dots+|a_n|^p\geq 1\}$ Of course $m_1$ is finite, since the series diverges.
$m_2:=\min\{n>m_1: |a_{m_1+1}|^p+|a_{m_1+2}|^p+\dots+|a_n|^p\geq1\}$ Again, the series minus a finite number of terms diverges, hence $m_2$ is finite.
...
$m_k:=\min\{n>m_{k-1}: |a_{m_{k-1}+1}|^p+\dots+|a_n|^p\geq 1\}$. Again $m_k$ is finite.
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We now create the following sequence of measurable sets:
$E_1:=[0,|a_1|^p]\times[0,1]^{d-1}$,
$E_2:=[|a_1|^p,|a_1|^p+|a_2|^p]\times[0,1]^{d-1},$
...
$E_{m_1}:=[|a_1|^p+\dots+|a_{m_1-1}|^p, |a_1|^p+\dots+|a_{m_1}|^p]\times[0,1]^{d-1}$
$E_{m_1+1}:=[0, |a_{m_1+1}|^p]\times[0,1]^{d-1}$
$E_{m_1+2}:=[|a_{m_1+1}|^p, |a_{m_1+1}|^p+|a_{m_1+2}|^p]\times[0,1]^{d-1}$
...
$E_{m_2}:=[|a_{m_1+1}|^p+\dots+|a_{m_2-1}|^p, |a_{m_1+1}|^p+\dots+|a_{m_2}|^p]\times[0,1]^{d-1}$
$E_{m_2+1}:=[0,|a_{m_2+1}|^p]\times[0,1]^{d-1}$
you get the point. Define $f_n=\chi_{E_n}$. Then it is $\|f_n\|_p=\lambda_d(E_n)^{1/p}$ but $\lambda_d(E_n)=|a_n|^p$, therefore $\|f_n\|_p=|a_n|$. Now since $(a_n)\in\text{gr}^p(\lambda_d)$, we must have $f_n\to0$ a.e. on $\mathbb{R}^d$. This is of course not true: for any point $x$ of $[0,1]^d$ there are infinite terms of $(f_n)$ with $f_n(x)=1$, since the sets $E_{m_k+1},\dots, E_{m_{k+1}}$ cover $[0,1]^d$ for any $k$.