I'm taking an intro to probability class, and my book lists this as an axiom:
$$P(\bigcup_{k=1}^{\infty}A_k) = \sum_{k=1}^{\infty}P(A_k)$$
where $A_k$ is an event and $P(A_k)$ is the probability of $A_k$.
Now consider the question of picking a random number in an interval. For instance, a random number is picked between 0 and 1. What's the probability of picking 0.5?
I see two possible answers:
1) 0: If it's 0, then you get weird results. For instance, say you pick a random number between 0 and 1. What's the probability of the number being less than 0.5? Well using the axiom at the very top, the probability of this is the sum of the probabilities of picking each number less than 0.5. Since they are all 0, the probability of picking a number less than 0.5 is 0. Which is obviously not true.
2) Infinitesimals. This would give you a result I'm assuming would make sense
So the way I see it, probability for continuous sample spaces cannot exist without infinitesimals.
Am I misunderstanding something or can probability for continuous sample spaces not exist without infinitesimals?
For a uniform distribution on an interval, the probability of occurrence of any one point is 0. There is no need for infintessimals. Results you get are internally consistent. The probability of occurrence in any interval is proportional to the length of that interval.