I think I understand what the axiom is doing. I'm not sure it's strong enough, though. There is no guarantee that there is any N that is >= M_i for all i (or for all large enough i, a weaker version which I think is what is needed), nor an N that is <= them. But suppose there are such an upper Nu and a lower Nl, thus giving a continuous range between them of Np = p Nl + (1-p) Nu for all p in 0..1. There is no guarantee that the supremum of those p for which Np is a lower bound is equal to the infimum of those for which it is an upper bound. The axiom needs to stipulate that lower and upper bounds Nl and Nu exist, and that there is no gap in the behaviours of the family Np.
One also needs some axioms to the effect that a formal infinite sum Sum{i>=0: pi Mi} actually behaves like one, otherwise "Sum" is just a suggestively named but uninterpreted symbol. Such axioms might be invariance under permutation, equivalence to a finite weighted average when only finitely many pi are nonzero, and distribution of the mixture process to the components for infinite lotteries having the same sequence of component lotteries. I'm not sure that this is yet strong enough.
The task these axioms have to perform is to uniquely extend the preference relation from finite lotteries to infinite lotteries. It may be possible to do that, but having thought for a while and not come up with a suitable set of axioms, I looked for a counterexample.
Consider the situation in which there is exactly one sure-thing lottery M. The infinite lotteries, with the axioms I suggested in the second paragraph, can be identified with the probability distributions over the non-negative integers, and they are equivalent when they are permutations of each other. All of the distributions with finite support (call these the finite lotteries) are equivalent to M, and must be assigned the same utility, call it u. Take any distribution with infinite support, and assign it an arbitrary utility v. This determines the utility of all lotteries that are weighted averages of that one with M. But that won't cover all lotteries yet. Take another one and give it an arbitrary utility w. This determines the utility of some more lotteries. And so on. I don't think any inconsistency is going to arise. This allows for infinitely many different preference orderings, and hence infinitely many different utility functions.
The construction is somewhat analogous to constructing an additive function from reals to reals, i.e. one satisfying f(a+b) = f(a) + f(b). The only continuous additive functions are multiplication by a constant, but there are infinitely many non-continuous additive functions.
An alternative approach would be to first take any preference ordering consistent with the axioms, then use the VNM axioms to construct a utility function for that preference ordering, and then to impose an axiom about the behaviour of that utility function, because once we have utilities it's easy to talk about limits. The most straightforward such axiom would be to stipulate that U( Sum{i>=0: pi Mi} ) = Sum{i>=0: pi U(Mi)}, where the sum on the right hand side is an ordinary infinite sum of real numbers. The axiom would require this to converge.
This axiom has the immediate consequence that utilities are bounded, for if they were not, then for any probability distribution {i>=0: pi} with infinite support, one could choose a sequence of lotteries whose utilities grew fast enough that Sum{i>=0: pi U(Mi)} would fail to converge.
Personally, I am not convinced that bounded utility is the way to go to avoid Pascal's Mugging, because I see no principled way to choose the bound. The larger you make it, the more Muggings you are vulnerable to, but the smaller you make it, the more low-hanging fruit you will ignore: substantial chances of stupendous rewards.
In one of Eliezer's talks, he makes a point about how bad an existential risk to humanity is. It must be measured not by the number of people who die in it when it happens, but the loss of a potentially enormous future of humanity spreading to the stars. That is the real difference between "only" 1 billion of us dying, and all 7 billion. If you are moved by this argument, you must see a substantial gap between the welfare of 7 billion people and that of however many 10^n you foresee if we avoid these risks. That already gives substantial headroom for Muggings.
I think I understand what the axiom is doing. I'm not sure it's strong enough, though. There is no guarantee that there is any N that is >= M_i for all i (or for all large enough i, a weaker version which I think is what is needed), nor an N that is <= them.
The M_i's can themselves be lotteries. The idea is to group events into finite lotteries so that the M_i's are >= N.
Personally, I am not convinced that bounded utility is the way to go to avoid Pascal's Mugging, because I see no principled way to choose the bound.
There is no principled ...
Summary: the problem with Pascal's Mugging arguments is that, intuitively, some probabilities are just too small to care about. There might be a principled reason for ignoring some probabilities, namely that they violate an implicit assumption behind expected utility theory. This suggests a possible approach for formally defining a "probability small enough to ignore", though there's still a bit of arbitrariness in it.