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In complex analysis, sometimes we need to use some theorems which are results of measure theory. However, I know very very little about measure theory. So

What are some very basic results of measure theory on complex functions/complex plane/complex calculus?

I expect the answers to be like:

  1. ... is always measurable.
  2. ... is always integrable.
  3. For theorems considering a measure space, we can choose it to be ...
  4. ...(anything that is worth mentioning)

For instance, I always suspected all harmonic functions are measurable, but don’t know how to prove it.

Another example is, I failed to apply Dominated convergence theorem rigorously. On the Wikipedia page, we need to consider $\{f_n\}$ a sequence of measurable real functions on a measure space $(S,\Sigma,\mu)$. What $\{f_n\}$ can I choose if the function is complex? What measure space should I consider?

I hope I have provided enough context and my question is not too broad.

Thanks for any help in advance.

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Continuous functions are measurable. All the single-valued functions you'll see in complex variables are measurable. In particular, harmonic functions are measurable.

As for convergence theorems, I can't think of any but the dominated convergence theorem that are likely to apply. (Way back when I took complex variables, we used Alfohrs, which doesn't use measure theory, so I'm guessing, but I think this is right.)

As for what sequences of functions to pick, you probably want a sequence of analytic functions. To apply the dominated convergence theorem, it would be enough that they are bounded in modulus by an integrable function. That is, $|f_n(z)|\le |f(z)| $ where $f_n\to f$ pointwise, and $f$ is integrable on the domain in question. Or you could apply the theorem to the real and imaginary parts separately.

The measure space will be the domain of the functions, with Lebesgue measure. You don't have to worry about that too much.

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  • $\begingroup$ So I don’t need to worry about that ‘$\sigma$-algebra’? $\endgroup$ – Szeto Jul 1 '18 at 1:13
  • $\begingroup$ No you don't need to worry about that for now. The $\sigma-$algebra stuff is used for proving theorems about measure theory. You don't need it to apply the theorems, once you know you've got a measure space. In the plane there's no problem. (It's hard to prove that there are any subsets of the plane that are not Lebesgue-measureable and impossible to construct one explicitly.) $\endgroup$ – saulspatz Jul 1 '18 at 1:17
  • $\begingroup$ Whoops, my last comment suggests that you're dealing with two-dimensional Lebesgue measure, but you're really not, because you're always dealing with line integrals. However, the thrust of it is right. You don't need to worry about $\sigma-$algebras. $\endgroup$ – saulspatz Jul 1 '18 at 1:26
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I would suggest you take a look at Folland : Real Analysis and Modern Techniques. The chapter 0 can be ommited and simply used as reference when you forget some basic analysis results. Chapter 1 is a bit harsh at the beginning so maybe skim through it to have a general idea of what is a measure but the most interesting part for you is chapter 2 which is really well explained and has everything (almost) you need about measurability of functions and integration of functions. It treats everything in a general setting and emphasize real and complex measure spaces. Good luck !

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