Simple P-Integrable Functions Dense in Lebesgue Space

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Theorem

Let $\left({X, \Sigma, \mu}\right)$ be a measure space, and let $p \in \R$, $p \ge 1$.

Let $\mathcal{L}^p \left({\mu}\right)$ be Lebesgue $p$-space for $\mu$.

Let $\mathcal E \left({\Sigma}\right) \cap \mathcal{L}^p \left({\mu}\right)$ be the space of $\Sigma$-simple, $p$-integrable functions.


Then $\mathcal E \left({\Sigma}\right) \cap \mathcal{L}^p \left({\mu}\right)$ is everywhere dense in $\mathcal{L}^p \left({\mu}\right)$.


Proof

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