Defining an unusual subspace of $c_0$ This is going to be a long post, so I'm giving a description first:
I recently came across the following exercise: Let $(X,\mathcal{A},\mu)$ be a measure space. If $(f_n)\subset L^p(\mu)$ and $f\in L^p(\mu)$, $p\geq 1$, such that $\|f_n-f\|_p\leq n^{-c}$ with $c>1/p$, prove that $f_n\to f$ a.e. on $X$.  The solution is not a big deal, but leads to an interesting question: which rates are good enough for convergence in norm to imply convergence a.e.? Recall that with no further assumptions, convergence in norm only implies the existence of a subsequence that converges a.e.
Anyway, the following definition is necessary: Define $\text{gr}^p(\mu)$ as the set $$gr^p(\mu):=\{(a_n)\in c_0|\text{ for all } (f_n)\subset L^p(\mu): \big{(}\forall n\in\mathbb{N}: \|f_n\|_p\leq |a_n|\big{)}\implies f_n\to0\text{ a.e.}\}$$
What I would like is to describe this set. My progress is the following:
1) For any measure space and any $p\geq 1$ it is $(0)\in\text{gr}^p(\mu)$, therefore this set is never empty.
2) If $(a_n)\in\text{gr}^p(\mu)$ and $\lambda\in\mathbb{C}$ then $\lambda\cdot(a_n)\in\text{gr}^p(\mu)$.
Indeed, if $\lambda=0$ it is obvious; otherwise if $(f_n)\subset L^p(\mu)$ with $\|f_n\|_p\leq|\lambda a_n|$ for all $n$ we have that $\displaystyle{\|\frac{1}{\lambda}f_n\|_p\leq|a_n|}$ for all $n$ therefore $\frac{1}{\lambda}f_n\to 0$ a.e. which is true iff $f_n\to 0$ a.e.
3) For any measure space, $\ell^p\subset\text{gr}^p(\mu)$.
Let $(a_n)\in\ell^p$ and $(f_n)\subset L^p(\mu)$ with $\|f_n\|_p\leq|a_n|$ for all $n$. We have $\displaystyle{\int_X|f_n|^pd\mu\leq|a_n|^p}$ for all $n$ and by summing and using the Monotone convergence theorem we have that $\displaystyle{\int_X\sum_{n}|f_n|^pd\mu\leq\|(a_n)\|_{\ell^p}<\infty}$, therefore the series $\sum_n|f_n|^p$ converges a.e. hence $|f_n|^p\to0$ a.e. which implies $f_n\to 0$ a.e.
4) In any measure space, if $(a_n)\in\text{gr}^p(\mu)$ and $(a_{n_k})\subset(a_n)$, we have $(a_{n_k})\in\text{gr}^p(\mu)$.
Let $(a_{n_k})\subset(a_n)\in\text{gr}^p(\mu)$ and $(f_k)\subset L^p(\mu)$ s.t. for all $k$ it is $\|f_k\|_p\leq |a_{n_k}|$; Define $g_n$ as $0$ if $n\not\in\{n_k: k\in\mathbb{N}\}$ and $g_{n_k}=f_k$ for all $k$. Then $\|g_n\|_p\leq |a_n|$ for all $n$, hence $g_n\to 0$ a.e. which of course implies $f_k\to0$ a.e.
5) $\text{gr}^p(\mu)$ is a linear subspace of $c_0$.
We need only to prove that it is closed under addition. Let $(a_n),(b_n)\in\text{gr}^p(\mu)$ and $(f_n)\subset L^p(\mu)$ with $\|f_n\|_p\leq|a_n+b_n|$ for all $n$. We partition $\mathbb{N}$ in $S=\{n: a_n=0\}$ and its complement $\mathbb{N}-S$.
Case 1: $S$ is an infinite set. By 4), $(b_n)_{n\in S}\in\text{gr}^p(\mu)$ and $\|f_n\|_p\leq|b_n|$ for all $n$ in $S$; therefore the subsequence $(f_n)_{n\in S}$ converges a.e. to $0$. Now for $n\not\in S$ we can find $\lambda_n\in\mathbb{C}$ such that $b_n=\lambda_n\cdot a_n$. We have to deal with two sub-cases:
Sub-case 1: There exists $M>0$ s.t. for all $n\in\mathbb{N}-S$ it is $|\lambda_n|\leq M$.
In this sub-case, for $n\not\in S$ we have $\|f_n\|_p\leq |a_n|+|b_n|\leq (1+M)|b_n|$. But $(b_n)_{n\in\mathbb{N}-S}\in\text{gr}^p(\mu)$ by 4), and by 2) we have $((1+M)b_n)_{n\in\mathbb{N}-S}\in\text{gr}^p(\mu)$. Hence $(f_n)_{n\in\mathbb{N}-S}$ converges to $0$ a.e.
Sub-case 2: $|\lambda_n|\to\infty$ as $n\to\infty$ through $\mathbb{N}-S$ (note that if $\mathbb{N}-S$ is finite we are automatically in sub-case 1).
We can find $n_0\in\mathbb{N}$ such that for all $n\geq n_0$ and $n\not\in S$ it is $|\lambda_n|>1$. For those $n$ it is $a_n=\frac{1}{\lambda_n}b_n$ therefore $\|f_n\|_p\leq|1+1/\lambda_n|\cdot|b_n|\leq2|b_n|$; now since $(b_n)_{n\geq n_0, n\in\mathbb{N}-S}\in\text{gr}^p(\mu)$ it is $(f_n)_{n\geq n_0, n\in\mathbb{N}-S}\to0$ a.e. and we are done.
Case 2: S is finite; we can do exactly what we did in the two sub-cases above for $\mathbb{N}-S$ and we are done.
Anyway, my questions to the community are these:

*

*Are these spaces any interesting in your opinion?


*What would be a good norm for these spaces? I can't think of anything that is of interest.
In this post I prove that for a series of Dirac point-mass measures the space $\text{gr}^p(\mu)$ is the entire $c_0$ for all $p$ and that for the measure space $(\mathbb{R}^d, \mathcal{L}^d, \lambda_d)$ the space $\text{gr}^p(\mu)$ is only $\ell^p$.
EDIT: An easy argument that shows that $\text{gr}^p(\mu)$ is not closed under the supremum norm, unless $\text{gr}^p(\mu)=c_0$: obviously $c_{00}\subset\text{gr}^p(\mu)$, where $c_{00}$ denotes the subspace of sequences that are eventually $0$. Thus $\text{gr}^p(\mu)$ is dense in $c_0$ with the supremum norm.
 A: Per your profile, this is so you can sleep at night.
Your space is exactly $\ell^p$. Here is a proof:
Suppose that $\|a\|_p<\infty$ and $\|f_n\|_p\le |a_n|$ for each $n$. Then $\|f_n^p\|_1\le |a_n|^p$ for each $n$, so that $\|\sum |f_n|^p\|_1<\infty$. In particular, $\sum_n |f_n|^p$ is finite almost everywhere, so that $|f_n|^p$ converges to 0 almost everywhere and hence $f_n$ converges to 0 almost everywhere.
For the converse, suppose that $\|a\|_p=\infty$. Then we construct a sequence of functions $f_n$ on $[0,1]$ with $\|f_n\|_p\le a_n$ so that $f_n(x)$ does not converge for every $x\in [0,1]$. Let $b_n=\min(|a_n|^p,\frac 12)$ and notice that if $f_n$ is the indictator function of an interval of length $b_n$, then $\|f_n\|_p=b_n^{1/p}\le |a_n|$. By assumption, $\sum b_n=\infty$.
Now we just use the standard construction of a sequence of functions that converges to zero in $L^1$, but not pointwise. Let $t_0=0$ and $t_{n}=(t_{n-1}+b_n)\bmod 1$ for each $n$. Then let $f_n=\mathbf 1_{[t_{n-1},t_n]}$ (where if $t_n<t_{n-1}$, this means $\mathbf 1_{[t_{n-1},1]} + \mathbf 1_{[0,t_n]}$. Now it is known that $\limsup f_n(x)=1$ for all $x$ and $\liminf f_n(x)=0$ for all $x$. (This has been described to me as the "Goodyear blimp": the support of the function continually moves to the right, each one starting where the previous one finished. The blimp passes overhead infinitely often because the $b_n$ are not summable).
