Hahn-Vitali-Saks theorem: do we need to ask for finite total variations?

Question

In Yosida's functional analysis (p70), we encounter the Hahn-Vitali-Saks theorem, as also stated on wikipedia:

https://en.wikipedia.org/wiki/Vitali%E2%80%93Hahn%E2%80%93Saks_theorem

In particular, one of the conditions is that for all $n \geq 1$, the complex measures $\lambda_n$ must have finite total variations $|\lambda_n|(S)$.

However, in Rudin's book Real and Complex analysis, in theorem 6.4 (p119) we read that the total variation of a complex measure is always finite (it is even bounded!)

Does this mean that we can drop the condition that the complex measures $\lambda_n$ must have finite total variations in the formulation of the Hahn-Vitali-Saks theorem, as this condition is always satisfied? Or am I missing something?

Answer 1

When Rudin defines "complex measure" he requires the measure to be finite everywhere. That excludes many well-behaved measures, including Lebesgue measure.

I think it's also common to use "complex measure" to simply emphasize that the measure is not necessarily positive, but it might be allowed to include infinite measure cases.