TITLE: What are all measurable maps $f:\mathbb C\to\mathbb C$ such that $f(ab)=f(a)f(b)$?
QUESTION [20 upvotes]: Is there a nice description of all nonzero measurable functions $f:\mathbb{C}\to\mathbb{C}$ such that $f(ab)=f(a)f(b)\ $ for all $a$ and $b$ in $\mathbb{C}$?   
This is inspired by the question Multiplicative Analytic Functions, Theo Buehler's comments there, and idle curiosity.   The only examples I know have the form $f(z)=|z|^cz^k$ or $f(z)=|z|^c\overline z^k$ for some $c\in[0,\infty)$ and nonnegative integer $k$.  A subquestion is: Are these all of them? (Answer: No, see update.)
I tried searching, and found that nonzero multiplicative maps from a semigroup to the multiplicative semigroup of complex numbers are sometimes called semicharacters, and other times called characters, but so far this hasn't helped me find anything that answers this question.
Update: I found from reading a remark in Grillet's Commutative semigroups that if $f$ is an example, then so is the function $g$ defined by $g(0)=0$, $g(z)=f(z)/|f(z)|$ if $z\neq 0$, and $g$ maps into the unit circle unioned with $\{0\}$.  This then made me realize that if $f$ and $g$ are examples, then so is the function $h$ defined by $h(0)=0$, $h(z)=f(z)/g(z)$ if $z\neq 0$.  So there are examples I did not include above.  I would have to include $f(0)=0$, $f(z)=|z|^cz^k$ where $c$ is any real number and $k$ any integer.
Also, I'm not even sure about what all of the continuous examples are.

REPLY [8 votes]: If you're willing to assume all such measurable $f(z)$ are continuous on ${\mathbb C} - \{0\}$ then it's doable. (I prove this for locally integrable $f(z)$ below). First observe $f(1) = f(1)f(1)$ so $f(1) = 0$ or $1$. If $f(1) = 0$, then $f(z) = f(z)f(1) = 0$ for all $z$ and there's not much to say. So assume $f(z) = 1$ in the following.
Next, Since $f(z^n) = f(z)^n$ for all $z$ and all integers $n$, it suffices to find the form of $f(z)$ in a neighborhood of $z = 1$. Equivalently, it suffices to find the form of $g(r,s) = f(e^re^{is})$ for all $(r,s)$ in a neighborhood of the origin. Since $g(0,0) = 1$ and we are assuming $f(z)$ to be continuous, there is a well-defined $h(r,s) = \log(g(r,s))$ on a neighborhood of the origin. The functional equation translates into $h(r_1 + r_2,s_1 + s_2) = h(r_1,s_1) + h(r_2,s_2)$, where as before the variables are near the origin. So the real and imaginary parts of $h(r,s)$ must also satisfy this functional equation.
Since $Re(h)$ and $Im(h)$ are continuous real-valued functions satisfying the functional equation, by the well-known additive case (whose proof works even for functions only defined in a neighborhood of the origin) we have that $Re(h)(r,s) = ar + bs$ and $Im(h)(r,s) = cr + ds$ for some real $a,b,c,$ and $d$. In terms of $g(r,s)$, this gives 
$$g(r,s) = e^{ar + bs}e^{icr + ids}$$
Not every $g(r,s)$ of this form will work globally; since $g(r,s) = f(e^re^{is})$ one must have that $g(r,s+2\pi) = g(r,s)$. This forces $b = 0$ and $d$ to be a multiple of $2\pi$. So we have $$f(e^re^{is}) = g(r,s) = e^{ar}e^{icr + ins}$$
Here $n$ is an integer. Equivalently where $m$ denotes $a - n$, we have (for $z \neq 0$):
$$f(z) = |z|^me^{ic\ln|z|}z^n$$
Conversely, since each of the factors $|z|^m$, $e^{ic\ln|z|}$ and $z^n$ satisfy the functional equation so do their product. Thus these are exactly the functions continuous on ${\mathbb C} - \{0\}$ satisfying the functional equation, other than the zero function.
Now back to why measurable examples are always continuous. Based on ccc's answer it seems there is a general theory that gives this. For locally integrable $f(z)$ it can be proven directly. Let $\phi(z)$ be a bump function with integral 1 supported on a neighborhood of the origin. Then for any $z$ and $w$ you have
$$f(z)f(w)\phi(w) = f(zw)\phi(w)$$
Integrate both sides with respect to $w$. You get
$$f(z) = \int f(zw)\phi(w)\,dw$$
Doing a change of variables this becomes
$$= {1 \over |z|^2}\int f(w)\phi(z^{-1}w)\,dw$$
(We assume $z \neq 0$). By the dominated convergence theorem the right-hand side is continuous (even smooth) as a function of $z$, so that $f(z)$ is a continuous function.