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:
- … is always measurable.
- … is always integrable.
- For theorems considering a measure space, we can choose it to be …
- …(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.
Best Answer
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.