Does absolute continuity of measures imply a relation between the $L_p$ spaces?

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Say $(X,\mathcal{B},\mu)$ is some measure space, and let $\sigma$ be some other measure on $(X,\mathcal{B})$ such that $\sigma\ll\mu$. What can one say about the relation between $L_p(\mu)$ and $L_p(\sigma)$?