Consider the following fragment from Takesaki's first volume (chapter IV):
For context, $E$ is a Banach space, $\Gamma$ is a locally compact Hausdorff space and $\mu$ is a positive Radon measure on $\Gamma$. The Banach space $L_E^p(\Gamma, \mu)$ is formed as follows: we consider elements $C_c(\Gamma,E) = C_c(\Gamma)\otimes E$ where $\|f-g\|_p = 0$ identifies $f$ as $g$. We then consider the Banach space completion of this quotient space.
My question is: which version of Egoroff's theorem is Takesaki using? Every version of this theorem that I know at least assumes that the codomain of the sequence of functions consists of a separable metric space. In this case, the Banach space $E$ may not be separable so I don't know which version of Egoroff's theorem applies.
Thanks in advance!
Since every $(f_n)$ is compactly supported, its image is compact, hence separable. Thus the linear span of $\bigcup_n f_n(\Gamma)$ is separable, and you may replace $E$ by the closure of this space.