Htykuut

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 $fin L^p(mu)$, $pgeq 1$, such that $|f_n-f|_pleq n^{-c}$ with $c>1/p$, prove that $f_nto f$ 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 $$gr^p(mu):={(a_n)in c_0|text{ for all } (f_n)subset L^p(mu): big{(}forall ninmathbb{N}: |f_n|_pleq |a_n|big{)}implies f_nto0text{ a.e.}}$$

What I would like is to describe this set. My progress is the following:

1) For any measure space and any $pgeq 1$ it is $(0)intext{gr}^p(mu)$, therefore this set is never empty.

2) If $(a_n)intext{gr}^p(mu)$ and $lambdainmathbb{C}$ then $lambdacdot(a_n)intext{gr}^p(mu)$.

Indeed, if $lambda=0$ it is obvious; otherwise if $(f_n)subset L^p(mu)$ with $|f_n|_pleq|lambda a_n|$ for all $n$ we have that $displaystyle{|frac{1}{lambda}f_n|_pleq|a_n|}$ for all $n$ therefore $frac{1}{lambda}f_nto 0$ a.e. which is true iff $f_nto 0$ a.e.

3) For any measure space, $ell^psubsettext{gr}^p(mu)$.

Let $(a_n)inell^p$ and $(f_n)subset L^p(mu)$ with $|f_n|_pleq|a_n|$ for all $n$. We have $displaystyle{int_X|f_n|^pdmuleq|a_n|^p}$ for all $n$ and by summing and using the Monotone convergence theorem we have that $displaystyle{int_Xsum_{n}|f_n|^pdmuleq|(a_n)|_{ell^p}<infty}$, therefore the series $sum_n|f_n|^p$ converges a.e. hence $|f_n|^pto0$ a.e. which implies $f_nto 0$ a.e.

4) In any measure space, if $(a_n)intext{gr}^p(mu)$ and $(a_{n_k})subset(a_n)$, we have $(a_{n_k})intext{gr}^p(mu)$.

Let $(a_{n_k})subset(a_n)intext{gr}^p(mu)$ and $(f_k)subset L^p(mu)$ s.t. for all $k$ it is $|f_k|_pleq |a_{n_k}|$; Define $g_n$ as $0$ if $nnotin{n_k: kinmathbb{N}}$ and $g_{n_k}=f_k$ for all $k$. Then $|g_n|_pleq |a_n|$ for all $n$, hence $g_nto 0$ a.e. which of course implies $f_kto0$ 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 $(a_n),(b_n)intext{gr}^p(mu)$ and $(f_n)subset L^p(mu)$ with $|f_n|_pleq|a_n+b_n|$ 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), $(b_n)_{nin S}intext{gr}^p(mu)$ and $|f_n|_pleq|b_n|$ for all $n$ in $S$; therefore the subsequence $(f_n)_{nin S}$ converges a.e. to $0$. Now for $nnotin S$ we can find $lambda_ninmathbb{C}$ such that $b_n=lambda_ncdot a_n$. We have to deal with two sub-cases:

Sub-case 1: There exists $M>0$ s.t. for all $ninmathbb{N}-S$ it is $|lambda_n|leq M$.

In this sub-case, for $nnotin S$ we have $|f_n|_pleq |a_n|+|b_n|leq (1+M)|b_n|$. But $(b_n)_{ninmathbb{N}-S}intext{gr}^p(mu)$ by 4), and by 2) we have $((1+M)b_n)_{ninmathbb{N}-S}intext{gr}^p(mu)$. Hence $(f_n)_{ninmathbb{N}-S}$ converges to $0$ a.e.

Sub-case 2: $|lambda_n|toinfty$ as $ntoinfty$ through $mathbb{N}-S$ (note that if $mathbb{N}-S$ is finite we are automatically in sub-case 1).

We can find $n_0inmathbb{N}$ such that for all $ngeq n_0$ and $nnotin S$ it is $|lambda_n|>1$. For those $n$ it is $a_n=frac{1}{lambda_n}b_n$ therefore $|f_n|_pleq|1+1/lambda_n|cdot|b_n|leq2|b_n|$; now since $(b_n)_{ngeq n_0, ninmathbb{N}-S}intext{gr}^p(mu)$ it is $(f_n)_{ngeq n_0, ninmathbb{N}-S}to0$ 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 $ell^p$.