## Dense set in $L(\mu \times \mu)$ [closed]

Let μ be a finite (or σ-finite measure) measure on some σ-field S of subsets of a set X. Is it true that the set of all linear combinations of functions of the form $1_{E×F}$, where E,F∈S, $\mu(E),μ(F)<∞$, is dense in Banach space $L(\mu \times \mu)$) ?

Thanks

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By $L$ do you mean $L^2$ ? In this case this is not a research level question, see the FAQ for more appropriate forums to post it. Good luck – Adrien Hardy Aug 12 2011 at 8:34
In what context does your question arise? (That is, why are you considering it?) – Yemon Choi Aug 12 2011 at 9:11
It would be helpfuff for $L^2$, but I would want to know for $L^p$ with every $1\leq p \leq \infty$. – rts Aug 12 2011 at 9:27
It is well-known that the set of integrable simple functions is dense in $L^p$, with $1\leq p <\infty$. But in some approximations problems it would be easier to use characteristic functions of measurable rectangles than the characteristic functions of measurable subsets. – rts Aug 12 2011 at 9:37
This is a crosspost from math.SE: math.stackexchange.com/q/56496 It is the end of a series of about 10 posts on very basic measure theory over the span of about three days, so I imagine people got a bit tired of answering all those. – Theo Buehler Aug 12 2011 at 10:57
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