Definition 44.4.4 ($p$-Integrable).label Let $(M, \tau)$ be a tracial von Neumann algebra in standard form, $x \in \td M$, and $E: \cb_{[0, \infty)}\to M$ be the resolution of the identity for $|x|$, and $p \in [1, \infty)$, then $x$ is $p$-integrable if
\[\norm{x}_{L^p(M, \tau)}^{p} = \int_{[0, \infty)}\lambda^{p} \tau(E(d\lambda)) = \int_{[0, \infty)}\lambda^{p} dE_{\hat 1_M, \hat 1_M}< \infty\]
The space $L^{p}(M, \tau) = \bracsn{x \in \td M|\ \norm{x}_{L^p(M, \tau)}}$ equipped with the $p$-norm $\norm{\cdot}_{L^p(M, \tau)}$ is the noncommutative $L^{p}$ space associated with $(M, \tau)$.
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