The continuity property of probability

[Appleton]

If (E_{n})) is either an increasing or decreasing sequence of events, then
lim n\rightarrow∞ P(E_{n}) = P(lim n\rightarrow∞ (E_{n}))

This seems to be saying that the limit as n goes to infinity of the probability of an increasing or decreasing sequence of events is equal to the probability as n goes to infinity of an increasing or decreasing sequence of events. I can't see a significant difference that merits the kind of proofs I see in the textbooks. What is the significance of moving the limit inside the brackets? Clearly I'm missing something. Could someone give me some intuition on this please?

 

[Vargo]

There are different ways to answer this.

1. Mathematically, the two sides of the equation represent different operations, so a proof is required to show that they give you the same thing. On the left, you are looking at a limit of probabilities. On the right you are looking at one probability evaluated on a set that is a limit of sets. This equation is not one of the basic assumptions of probability theory; therefore, it demands proof.

2. Why intuitively? Suppose the sequence is decreasing down to the empty set. In that case, the right hand side is the probability of the empty set which is 0. However, the left side represents a limit of probabilities of nonempty events (i.e. positive probabilities). How do you know that there is not some kernel of positive probability that lies inside every nonempty set E_n and that prevents the probability of E_n from converging down to 0. That is more or less what the proof is addressing.

 

[Stephen Tashi]

. I can't see a significant difference that merits the kind of proofs I see in the textbooks.

Suppose \{E_n\} is a sequence of events (not necessarily an increasing or decreasing sequence of events). Is the result necessarily true? If not, then something needs to be proven about why the special case of an increasing or decreasing sequences of events implies the result.