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Question: if and then there exists a simple measurable function...

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If f E \small L_{1}(X, X, \mu ) and \small \varepsilon > 0 , then there exists a simple \small X -measurable function \small \varphi such that \small \left \| f -\varphi \right \|_{1} < \varepsilon . Extend this to \small L_{p} , \small 1\leq p < \infty . Is this true for \small L_{\infty } ?

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