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:

1) Are these spaces any interesting in your opinion?

2) 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 $$\ell^p$$.

• PS: $gr^p$ is short for "p-good rates", i didnt know what to call this – JustDroppedIn Nov 20 '18 at 21:49