If you have a measure on a countable set, lets number it 0, 1, 2, .. then you must have: m(i) >= 0 (since it's a measure).

And must also have

1 = m(0) + m(1) + ... (because it's a measure)

so

1 = Lim S(i)

Where S(i) is the partial sum going from 0 to i.

But if each m(i) = 0, then each partial sum is zero.

So 1 = Lim 0 = 0

It seems to me, the question is, how do we assign probabilities to the existence of life. One way I can imagine is the following. We think of the universe as a classical system, then there is a phase space for the entire universe. Now we can look at each trajectory through phase space and classify it as either having or not having life at at least one point. Then we can obtain the measure of the set of trajectories classified as having life. With this view it seems at least possible that life could have measure zero even though it does not seem likely to me and there might even be [non-]obvious reasons why the set could not have measure zero. I am not sure how the argument would change if one would try something similar but with a quantum mechanical instead of a classical description of the universe.

EDIT: Additional thought and I might be totally wrong because of a lack of mathematical understanding. Pick a point on a trajectory classified as containing life and perturb it in a way such that it only affects parts of the universe far away from life. Then all trajectories through the perturbed points would also still be classified as containing life. But I think the resulting set of trajectories would still have measure zero because we allowed only perturbation far away from life.

So to grow a single trajectory classified as containing life into a set of trajectories classified as containing life of non-zero measure would require being able to pick a point on the trajectory and perturb it in all dimensions and still have all perturbed trajectories classified as containing life. Seems possible but not obviously so to me.