Definition of positive measure
Why Rudin assumes that $\mu(A)<\infty$ for at least one $A\in \mathfrak{M}$? What about if $\mu(A)=\infty$ for any $A\in\mathfrak{M}$?
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$\begingroup$If $\mu (A)=\infty $ for all $A $, then $\mu $ doesn't distinguish between sets at all. You would have $\int f=\infty $ for all $f\geq0$ that are nonzero on at least one point, and you cannot even define the integral of any function that changes signs. So basically you get no measure theory whatsoever.
$\endgroup$ 7 $\begingroup$The definition of measure you quote doesn't give the full form of countable additivity, which would apply to a disjoint collection of cardinality zero the same as any finite or countably infinite cardinality.
Since an empty union is the empty set and an empty sum is zero, the full version of additivity would imply $\mu(\varnothing) = 0$.
The hypothesis that $\mu(\varnothing) = 0$ is equivalent (given additivity over countably infinite disjoint unions) to the hypothesis that there is at least one measurable set of finite measure.
Consequently, IMO, the intent is to have the full version of countable additivity while only giving a simplified statement of that property.
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