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 $\ell^p$.

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

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