How to understand Bass 3.7

Question

I was looking at exercise 3.7 on page 20 of: http://bass.math.uconn.edu/3rd.pdf

The notation $A \uparrow$ is defined to be a sequence of sets $A_1 \subset A_2 \subset ... $

What I don't understand is the question is claiming that $\mu_1(A) \subset \mu_2(A) \subset \mu_3(A) ... $ but $\mu_k$ is a function that maps sets to numbers, so what does it mean for the positive real number of $\mu_1$ to be contained in $\mu_2$?

Answer 1

The notation $\uparrow$ in general means that it in increasing with respect to the order in a partially ordered set.

In the $\sigma$-algebra $\mathcal{A}$ this order is $\subset$ (non-strict subset), so $A_i \uparrow$ means that $A_1 \subset A_2 \subset \ldots$ as they explained in 1.1 as you already noted.

In the space of real numbers $\mathbb{R}$ this order is $\leq$, so $\mu_n(A) \uparrow$ means that $\mu_1(A)\leq \mu_2(A)\leq \ldots$: I think it would be nice if they would at least explain this in 1.1.

Answer 2

I think this notation means that the sequence $$\mu_1 (A), \mu_2 (A), \ldots$$ is increasing.