↑ and ↓ relations on measures (Durett)

Question

In Rick Durrett's fourth edition of "Probability: Theory and Examples", an ↑ relation is defined on Theorem 1.1.1's sections (iii) and (iv):

As stated, $\uparrow$ and $\downarrow$ are defined as:

$$a_i \uparrow a := (a_1 \subset a_2\subset...) \land (\cup_ia_i = a)$$ $$a_i \downarrow a := (a_1 \supset a_2\supset...) \land (\cap_ia_i = a)$$

As $\subset$, $\supset$, $\cup$ and $\cap$ are operations between sets, $a_i$ and $a$ seem to be necessarily sets. Yet, in sections $(iii)$ and $(iv)$, formulas $\mu(A_i) \uparrow \mu(A)$ and $\mu(A_i) \downarrow \mu(A)$ are stated. $\mu(A_i)$ and $\mu(A)$ are measures, and, by the definition of measure, set functions whose image are real numbers:

As $\subset$, $\supset$, $\cup$ and $\cap$ cannot have numbers as arguments, how can $\mu(A_i) \uparrow \mu(A)$ and $\mu(A_i) \downarrow \mu(A)$ be?

Answer 1

$\uparrow$ and $\downarrow$ for sequence of real numbers have different context; $a_n\uparrow a$ for a real seqence $a_n$ and a real number $a$ means $a_n$ is increasing and converges to $a$. $\downarrow$ also have similar meaning, by just substitute the word "increasing" to "decreasing".

However, we can say that $\uparrow$ for set sequences and real sequences share same background. In both cases, $\uparrow$ means left-hand-side is increasing and converges to the right-hand-side.

Answer 2

$\mu(A_{i})\uparrow\mu(A)$ means that $\mu(A_{n})\leq\mu(A_{n+1})$ as numbers, and $\lim_{n}\mu(A_{n})=\sup_{n}\mu(A_{n})$.

In a sense, measure preserves ordering as a continuous map: $A\subseteq B$ implies $\mu(A)\leq\mu(B)$, and $A_{1}\subseteq A_{2}\subseteq\cdots$ implies $\mu(A_{1})\leq\mu(A_{2})\leq\cdots$ and the "limit point" of $A_{i}$, which is $\displaystyle\bigcup_{n}A_{n}$ is mapped to $\sup_{n}\mu(A_{n})$.