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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](http://math.stackexchange.com/q/223182/8271) question.2012-10-31
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    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?2012-10-31

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