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In the text of Real Analysis by Folland, he defines the total variation of a complex measure $\nu $ as the unique measure $|\nu|$ such that if $d\nu = f d\mu $, with $ f$ a $ \mu -$ integrable function, then $d|\nu| = |f| d\mu $.

I've studied real analysis before by Rudin (real and complex analysis), and the definition that Rudin gives is said to be equivalent by and exercise in Folland's book. The propertie that characterizes the total variation is proved in Rudin's book, but it uses a lot of theory which Folland didn't define yet.

So, I'm trying to follow the lines of the Folland's text to show that the total variation is well defined. First, we show that there is a positive (and finite) measure that $d \nu=f d\mu$. We can take $\mu $ as the total variation of the real and imaginary parts of $\nu$. Then, we can use the Lebesgue-Radon-Nikodym theorem, since we have just finite masures.

Now we have to check that if $ d \nu =f_1 d\mu_1= f_2 d\mu_2 $, $ f_j$ a $ \mu_j -$ integrable function, then $$|f_1|d\mu_1= |f_2|d\mu_2.$$

To see this, he takes $\lambda = \mu_1+\mu_2 $ and then apply the chain rule. I can't see why we can take, for example, $ d\mu_1 / d\lambda $. Is the measure $\lambda$ a $\sigma-$ finite measure? If it's not, how can we work with this derivates?

Integral
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prnchautmc
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  • The above answer is incorrect. I was also wondering the same question as OP. Does anyone have an answer although it is late? – Jacob Bak Jun 02 '20 at 13:27

1 Answers1

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It is a bit late but i just see this while i was looking something different. now, note that positive measure is complex measure only if it is finite. so both $\mu_{1}$ and $\mu_{2}$ are finite positive measures. $\lambda = \mu_{1} + \mu_{2} $ would be finite positive measure.

Airbag
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