(Rudin's Functional Analysis 3.29 pg 81): Suppose $Q$ is a compact Hausdorff space, $X$ is a Banach space, $f:Q \rightarrow X$ is continuos, and $\mu$ is a positive Borel measure on $Q$. Then $$ \Big| \Big| \int_Q f \, d\mu \Big| \Big| \le \int _Q ||f|| \,d \mu . $$
I am concerned with the existence of integral.
We know that the integral $\int_Q \,f d\mu $ exists when $\mu$ is a finite probability measure. (3.27, p78)
I see this means if $\mu$ is a finite positive measure, the integral exists by linearity.
How does this hold for general positive measures?
I am thinking we might need the fact that $\mu$ is somehow locally finite - but even that I do not know how to prove.