Summing infinitesimals to 1

Question

Tl;dr: Could someone give me an explanation as to how infinitesimal numbers might be added together to get 1?

Why I want to know: some people (philosophers) like to introduce infinitesimals into probability theory when considering cases like an infinite amount of coin tosses. To say that the event that this coin will land heads every time has probability 0, they think, is bad, as probability 0 should be reserved for the impossible. Instead, they say it should have an infinitesimal probability.

Fair enough. But presumably, the probability that the coin corresponds to some infinite sequence of heads and tails is 1. If we like countable additivity, this means the probability of all the possible infinite sequences should sum up to 1.

Apparently this is consistent with their theory. Each infinitesimal probability of each infinite sequence will, when added together, equal 1.

I'm confused as as to how this could happen. Why would an infinite amount of infinitesimals add up to 1, and not, say, 2?

Answer 1

Note that the geometric series $$1/2 + 1/4 +1/8+.... =\sum _1 ^\infty \frac {1}{2^n} =1$$

More interesting is for a small positive real number $\epsilon$ $$\sum _1 ^\infty \frac {\epsilon}{2^n} =\epsilon $$

Therefore you can add infinitely many "infinitesimals and get another infinitesimal.

Answer 2

The short answer is you need to integrate up the infinitesimals, rather than summing them. This is because infinitesimal probabilities emerge from continuous distributions. With a pdf $f(x)$, $f(x)dx$ is an infinitesimal probability for $x$ existing in a width-$dx$ range of values.

We can flesh this out. Take an infinite sequence of coin tosses and label the tosses with integers $\ge1$. If $S$ is the set of tosses yielding heads, associate the sequence with $x(S):=\sum_{n\in S}2^{-n}\in[0,\,1]$. In countably infinitely many cases, a number is achievable with multiple sequences; but integration doesn't care about that. For any subset $T$ of $[0,\,1]$ for which $\int_Tdx$ is defined, we can say that integral is the probability of the sequence's representation being an element of $T$. Unitarity is then the condition $\int_0^1dx(S)=1$.