Strange behaviors of finitely additive probabilities

Question

Watching a lecture on youtube I heard the lecturer stating that in general finitely additive probabilities behaves strangely. For example, it is possible that every open interval around a point $x$ has probability $1/2$, while $p(\{x\})=0$.

This lead me to various questions:

1. Is there somebody who can explain how do we get to this specific result?
2. What are other examples of strange behavior by finitely additive probabilities?
3. Why – in general – countably additive probabilities let us avoid these kind of strange situations?

As always, any feedback will be most welcome.
Thank you for your help and your time.

[Just to underline a point, being self-taught, I do rely occasionally on youtube videos, but I am very selective about the provenience of the videos. In this case, the lecturer was a top theorist from a top university]

Answer 1

With countable additivity, it is easy to prove continuity of measure from below and from above. That is, the measure of an increasing countable union is the limit of the measures of the finite unions. The same holds for decreasing intersections provided one of the sets has finite measure (which holds for probability of course). This would prevent your case, in that it would give $ P (\{ x \}) =1/2$.

Basically if you only assume finite additivity then you can't do limit processes of any sort.

Unfortunately, writing down an example of such an object on a "rich" $\sigma $-algebra like the Lebesgue measurable subsets of the reals is impossible, because their nonexistence is consistent with ZF.

Answer 2

The probability measure defined by the Kolmogorov axioms (which include countable additivity) are set-continuous: the probability of a limit is the limit of the probabilities. If one denies countable additivity and accepts only finite additivity, then this set-continuity property is no longer applicable.

Consider the following notion from Calculus 101 (which is perhaps describable as the study of properties of continuous functions). Suppose $f(x)$ is a function that approaches a limit $L$ as $x \to a$. Then, it is not necessary that $f(a)$ be equal to $L$; $f(a)$ can have any arbitrary value. But when $f(a)$ does equal $L$, we say that $f(x)$ is continuous at $a$. If $f(x)$ is continuous at all $a \in (-\infty,\infty)$, we call it a continuous function. Similarly, countable additivity gives us set-continuity for all events in the $\sigma$-algebra but finite additivity does not.