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Let $L_p$ be the complete, separable space with $p>0$. $\mathbf{J}=\{I = (r,s] \}$ where $r$ and $s$ are rational numbers. $\mathbf{A}$ is the algebra generated by $\mathbf{J}$, with $\mathbf{S}=\operatorname{span}(\mathbf{A})$.

a). Try to verify that $\mathbf{S}$ is dense in $L_p$ space with respect to $L_p$ metric.

b). Try to verify that for any $p>0$, $L_p$ is complete.

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This is very inaccurate. b) is asked on this question. –  leo Oct 31 '12 at 3:36
    
I closed the other question @leo linked to as a duplicate of this one. But as leo said, this question can use some clarification. Since notations are highly mutable and dependent on context, try to write in words in addition to the notations. In particular, I am guessing (but I am not sure) that $\mathbf{J}$ refers to the set of characteristic functions of half open intervals, which makes $\mathbf{S}$ the set of step functions? –  Willie Wong Oct 31 '12 at 9:03

1 Answer 1

  • By definition of Lebesgue integral, simple functions (i.e. element of the vector space generated by the characteristic functions of measurable sets) are dense in $L^p$, $p>0$. So we have to show that each characteristic function of a Borel set of finite measure can be approximated by an element of $\bf S$ for the $L^p$ norm. Using this result, and the fact that finite disjoint unions of elements of $\bf J$ form an algebra which generates the Borel $\sigma$-algebra, we are done.

  • For $0<p<1$, see this question. For $1\leq p\leq \infty$, see this one (it deals with the case the function takes their values on Banach spaces).

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