# Notation on Resnick's for proving continuity of P and Fatou's lemma

I'm using Resnick's "A probability path" and I'm bit confused with his notation (particularly regarding $$\uparrow$$ and $$\downarrow$$ )when proving the continuity of the measure P for monotone sequences (page 31) and when proving Fatou's lemma (page 32).

## 1. Continuity of P for monotone sequences

If $$A_n \uparrow A$$, where $$A_n \in \mathcal{B}$$, then $$P(A_n) \uparrow P(A)$$

In proposition 1.4.1. (page 8), the author talks about monotone sequences and defines $$A_n \uparrow$$ as indicating that $$A_n$$ is a monotone non-decreasing sequence. In that case, then we can define the limit of $$A_n \uparrow$$ as:

$$\lim_{n \rightarrow \infty} A_n = \cup_{n=1}^\infty A_n$$

So I can interpret $$A_n \uparrow A$$ as saying that the sequence $$A_n$$ is non-decreasing and $$A = \cup_{n=1}^\infty A_n$$.

My problem is I'm not sure what $$P(A_n) \uparrow P(A)$$ means. Is it a regular limit, but the $$\uparrow$$ indicates that $$P(A_n)$$ is non-decreasing? Or does it indicate a $$lim inf$$ or some other convergence concept?

Concretely, I'm confused by the following step:

First he constructs a disjoint sequence of events

$$B_1 = A_1, B_2 = A_2 \setminus A_1, ..., B_n = A_n \setminus A_{n-1}, ...$$

and so $$\cup_{i=1}^n B_i = A_n, \cup_{i=1}^n B_i = \cup_{i} A_i ) = A$$

And then:

$$P(A) = P(\cup_{i=1}^\infty P(B_i) B_i) = \sum_{i=1}^\infty P(B_i) = \lim_{n \rightarrow \infty} \uparrow \sum_{i=1}^n P(B_i)$$

I'm confused by that last equality and what $$\lim_{n \rightarrow \infty} \uparrow \sum_{i=1}^n P(B_i)$$ means. I'm guessing I can't say that

$$\sum_{i=1}^\infty P(B_i) = \lim_{n \rightarrow \infty} \sum_{i=1}^n P(B_i)$$

because then I would be able to conclude that $$P(A_n) \rightarrow P(A)$$, which seems stronger that what the property says, but I don't get why it wouldn't be true.

This answer might be relevant, though it doesn't use the $$\uparrow$$ notation

You can view $$P(A_n) \uparrow P(A)$$ as saying that $$P(A_n)$$ converges to $$P(A)$$ from below. Resnick writes $$P(A_n) \uparrow P(A)$$ as shorthand for saying both that

1. $$\lim_{n \to \infty} P(A_n) = P(A)$$, and
2. $$P(A_n) \leq P(A_{n+1})$$ (the sequence is non-decreasing).

In light of item 2 above, you can replace 1 by something like, $$P(A) = \sup_n P(A_n)$$ or $$P(A) = \liminf_{n \to \infty} P(A_n)$$, et cetera.

Similarly, the "$$\lim_{n\to\infty} \uparrow$$" notation signifies a limit from below. In the case of $$\sum_{i=1}^\infty P(B_i)$$, it's just saying that a series of non-negative terms is the limit of the non-decreasing sequence of its partial sums.

• Thanks! That cleared up stuff
– cd98
Commented Aug 26, 2020 at 16:44

I mostly agree with Artem's answer except for the comment that the $$\lim_{n\to\infty} \uparrow$$ notation signifies a limit from below. If this were true then $$\lim_{n\to\infty} \downarrow (\bigcup_{k \ge n} A_k)$$ (on page 33, in the proof of Fatou's lemma) would signify a limit from above, which does not make sense. It seems that the down arrow in my example is merely signifying that $$\{\bigcup_{k \ge n} A_k\}$$ is a monotone nonincreasing sequence and therefore the limit is well-defined using Proposition 1.4.1. It is a little superfluous and can be ignored. In the original question, $$\lim_{n \rightarrow \infty} \uparrow \sum_{i=1}^n P(B_i)$$ is the same as $$\lim_{n \rightarrow \infty} \sum_{i=1}^n P(B_i)$$. After all, $$\{\sum_{i=1}^n P(B_i)\}$$ is a monotone nondecreasing sequence that is also bounded (since $$B_1, B_2, \dotsc$$ are disjoint and the sum of the probabilities assigned to them is at most $$1$$) and the limit of such a sequence always exists using the monotone convergence theorem.