Um no. They're not well defined over the distribution, they will certainly be well defined over a finite population.
I'm lost. A statistical distribution characterizes a population (whether the population is an abstract construction or a literal concrete population); if the mean isn't well-defined for the population it oughtn't be well-defined for the distribution allegedly characterizing the population.
Taking annual income for concreteness, the support of a power law distribution would include, for example, $69 quadrillion. But no one actually earns so much (global economic activity, denominated in dollars, is simply too small), so the support of the actual annual income distribution must exclude $69 quadrillion. Consequently the actual annual income distribution and the power law distribution cannot actually be the same distribution; they have different support.
You seem to be confused about how distributions with infinite means work. Here's a good exercise: get some coins and flip them to obtain data in a St. Petersburg distribution notice that even though the distribution has infinite mean all your data points are still finite (and quite small).
In the case of the St. Petersburg distribution one defines an abstract data-generating process which, by construction, implies a particular distribution with infinite mean. In the case of people's incomes or wealth, by contrast, we know that the output of the data-generating process is constrained from above by the size of the economy, so the resulting population (and the distribution representing that population) must have finite mean income and finite mean wealth. (It's as if we were talking about an imperfect real-life instantiation of the St. Petersburg process where we knew the casino had a limited amount of money.)
Consequently the actual annual income distribution and the power law distribution cannot actually be the same distribution; they have different support.
Every actual population differs from a parameterised mathematical function with few parameters, and for pretty much anything you can measure, if the mathematical distribution has infinite support, there will be some reason that the population cannot. But the question to ask is not, are they different, but, does the difference make a difference?
The way to answer this question is to repeat the analysis in ...
Politics as gymnastics for rationalists. No one one Less Wrong is quite sure why politics is a taboo topic or how things got to be that way. What we do think we know is that politics is a great way to bring out the irrationality in people. So why not take advantage of that and use politics as a way to measure rationality? Since politics brings out the most irrationality, it should provide the strongest signal. Since there aren't useful objective metrics of how a political discussion went, we'd have to use subjective judgements by neutral third-party raters, kind of like they do in gymnastics. (In the comment thread for this post, feel free to find fights that you have no dog in, improvise a rationality rubric, and grade participants according to your rubric... let's see how it goes.)
Be a sheep. This is probably the exact opposite of what you were taught in your high school civics class. But if my friend Jane is more intelligent, more informed, and less ideological than I am, it seems like voting however Jane is going to vote is a strict improvement over voting however I would naively. It also saves me time, and gives Jane an incentive to put even more time in to carefully considering political issues since she now controls two votes instead of one. Done on a large scale, this could provide an interesting twist on representative democracy. Imagine a directed graph where each node represents a person and an edge is directed from person A to person B if person A is auto-copying person B's votes. There's a government computer system where you can change the person you're auto-copying votes from at any time or override an auto-copied vote with your own personal guess about what's best for society. Other than that, it's direct democracy... all bills are put before all citizens to vote on. Problems this might solve: