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

Question

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

Answer 1

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.

Answer 2

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.