I am reading Probability for Statistics and Machine Learning.
I have trouble understanding the arrow notation (similar to Knuth's arrow notation) in the following definition:
Theorem 1.1. Let $A_1 \supset A_2 \supset A_3 \supset \cdots$ be an infinite family of subsets of a sample space $\Omega$ such that $A_n \downarrow A$. Then, $P(A_n) \rightarrow P(A)$ as $n \rightarrow \infty$.
I have looked for this notation with relation to sets but I wasn't able to understand its meaning in this context.
Best Answer
It means “sequence of sets $A_n$ decreases to set $A$”: there exists set $A$ that is subset of all $A_n$. For example, $A_n = \left[-\frac{1}{n}, \frac{1}{n}\right]$ decreases to $A=\{0\}$.