How is countably infinite addition defined

Question

In the axiom of additivity of probability theory, the concept of a countably infinite sum, i. e. the sum of countably infinitely many real numbers, is used. Could someone please tell me how that kind of addition is defined?

Answer 1

The infinite sum is defined as$$\sum_{n=1}^{\infty}x_n = \lim_{n\rightarrow \infty}S_n $$ Where $$S_n = \sum_{k=1}^{n} x_k$$

Answer 2

It is the sum of a convergent series, i.e., the limit of the increasing sequence formed by the sums of the first $n$ terms as $n$ tends towards infinity (or $+\infty$ itself if no finite limit exists).

Answer 3

In probability all the addends are positive so the partial sums are monotonically increasing. The sum is bounded above by $1$, so the monotone convergence theorem applies and the sum converges.