Two questions on Banach function spaces

Question

I have recently started studying Banach function spaces over $\sigma$-finite measure spaces. By a Banach function space I mean:

Let $\left(R, \mu \right)$ be a $\sigma$-finite measure space and let $M$ detone the set of all measurable functions on $\left(R, \mu \right)$. We say that mapping $||\cdot||: M\rightarrow [0,\infty]$ is a Banach function norm if (for all $f_n, f, g\in M$ and $\alpha\in\mathbb R$):

1. $||f||=0 \Leftrightarrow f=0$ a.e., $||\alpha f||=|\alpha|||f||$ and $||f+g||\leq||f||+||g||$

2. $|f|\leq|g|\ a.e.\Rightarrow ||f||\leq||g||$

3. $0\leq f_n\uparrow f\Rightarrow ||f_n||\uparrow||f||$
4. For every measurable $E\subseteq R$, $\mu(E) < \infty$: $||\chi_E|| < \infty$
5. For every measurable $E\subseteq R$, $\mu(E) < \infty$ there exists a constant $C_E > 0$ (independent of $f$), such that $\int_E \! |f| \, \mathrm{d}\mu\leq C_E||f||$.

Let $X=\{f\in M;||f||<\infty\}$. Then $(X, ||\cdot||)$ is a Banach function space.

It's not hard to show that it is a Banach space, indeed. Clearly, $L^p$ spaces are Banach function spaces.

After having gone through the basics, I have two questions:

1) Are there some useful criteria for functions from a Banach function space to be essentially bounded?

2) Are there some useful criteria for a Banach function space to be closed under (pointwise) multiplication?

I've tried to think it up or google it but I found out nothing useful.

Thanks a lot for any help!:)

Answer 1

I will assume that you are interested in $L^p$ spaces, and not spaces of continuous functions. The answer to your questions depends mostly on the underlying measure space.

1. If $\Omega$ is an open subset of $\mathbb{R}^n$ with Lebesgue measure, then $L^p(\Omega)$ contains unbounded functions for all $p<\infty$. On the other hand all sequances in $\ell^p$ are bounded for all $p$.

2. Consider $L^p(0,1)$ with Lebesgue measure, $1\le p<\infty$. For all $a>0$, $x^{-1/p+a}\in L^p(0,1)$. However $(x^{-1/p+a})^2=x^{-2/p+2a}\in L^p(0,1)$ if and only if $a>1/(2\,p)$. This example is easily translated to $L^p(\Omega)$ where $\Omega$ is an open subset of $\mathbb{R}^n$. On the other hand, the product of two sequences in $\ell^p$ is in $\ell^p$ for all $p$.