I've started reading All of Statistics by Larry Wasserman with the hope of gaining understanding of machine learning fundamentals.
The author gives the following definition in the first chapter:
A sequence of sets $A_1, A_2,\dots$ is monotone increasing if $A_1
\subset A_2 \subset \dots$ and we define $\lim_{n\rightarrow
\infty}A_n = \bigcup_{i=1}^{\infty}A_i$.(The author then analogously defines the
monotone decreasing sequence of sets.)In either case, we will write $A_n \rightarrow A$.
It's obvious that for any finite k, $A_k = \bigcup_{i=1}^{k} A_k$, so I guess it's an infinite extension of this. My problem is I don't really understand the motivation behind this definition; i.e. what it "buys us".
I'd appreciate any insight.
Best Answer
Among other things, such constructions help you define and understand the continuity of measures (which is basically a function defined on sets in a sigma algebra). For example, with the $A_n$s you have defined, and with a measure $\mu$,
$$ \lim \mu (A_n)=\mu (\lim A_n)$$
Compare this with standard continuity below to see the similarities:
$$ \lim f (x_n)=f (\lim x_n)$$